Zombies have found out about the Zombie Contamination level checker and managed to damage it! Now detecting the shape of their main compound will be a real challenge for Heidi. As before, a lair can be represented as a strictly convex polygon on a lattice. Each vertex of the polygon occupies a point on the lattice. However, the damaged Zombie Contamination level checker can only tell, for each cell, whether the level of Zombie Contamination for that cell is in the set {1, 2, 3}. In other words, Heidi knows all the cells of the lattice for which the Contamination level is not 0 and not 4.Given this information, Heidi still wants to know the exact shape of the lair to rain destruction on the zombies. Help her!

The input contains multiple test cases. The first line of each test case contains two space-separated integers N and M, where N is the size of the lattice grid (5 ≤ N ≤ 100000) and M is the number of lattice points for which the Zombie Contamination level is 1, 2, or 3 (8 ≤ M ≤ 200000). The second line of each test case contains M pairs of integers x1, y1, ..., xM, yM – coordinates of the cells with Zombie Contamination level not equal to 0 nor 4. It is guaranteed that 1 ≤ xi, yi ≤ N. All pairs xi, yi are different. Cells are enumerated based on the coordinates of their upper right corner. This means that the bottommost leftmost cell that touches the origin has coordinates (1, 1), and the uppermost leftmost cell is identified as (1, N). The last line of the file contains two zeroes. This line should not be treated as a test case. The sum of the M values for all tests in one file will not exceed 200000.

For each test case, the following output is expected: The first line of the output should contain one integer V, the number of vertices of the polygon that is the secret lair. The next V lines each should contain two integers, denoting the vertices of the polygon in the clockwise order, starting from the lexicographically smallest vertex.

NoteIt is guaranteed that the solution always exists and is unique. It is guaranteed that in the correct solution the coordinates of the polygon vertices are between 1 and N - 1. A vertex (x1, y1) is lexicographically smaller than vertex (x2, y2) if x1 < x2 or .


"The zombies are lurking outside. Waiting. Moaning. And when they come...""When they come?""I hope the Wall is high enough."Zombie attacks have hit the Wall, our line of defense in the North. Its protection is failing, and cracks are showing. In places, gaps have appeared, splitting the wall into multiple segments. We call on you for help. Go forth and explore the wall! Report how many disconnected segments there are.The wall is a two-dimensional structure made of bricks. Each brick is one unit wide and one unit high. Bricks are stacked on top of each other to form columns that are up to R bricks high. Each brick is placed either on the ground or directly on top of another brick. Consecutive non-empty columns form a wall segment. The entire wall, all the segments and empty columns in-between, is C columns wide.

The first line of the input consists of two space-separated integers R and C, 1 ≤ R, C ≤ 100. The next R lines provide a description of the columns as follows:    each of the R lines contains a string of length C,  the c-th character of line r is B if there is a brick in column c and row R - r + 1, and . otherwise.  B

The number of wall segments in the input configuration.

NoteIn the first sample case, the 2nd and 3rd columns define the first wall segment, and the 5th column defines the second.


Further research on zombie thought processes yielded interesting results. As we know from the previous problem, the nervous system of a zombie consists of n brains and m brain connectors joining some pairs of brains together. It was observed that the intellectual abilities of a zombie depend mainly on the topology of its nervous system. More precisely, we define the distance between two brains u and v (1 ≤ u, v ≤ n) as the minimum number of brain connectors used when transmitting a thought between these two brains. The brain latency of a zombie is defined to be the maximum distance between any two of its brains. Researchers conjecture that the brain latency is the crucial parameter which determines how smart a given zombie is. Help them test this conjecture by writing a program to compute brain latencies of nervous systems.In this problem you may assume that any nervous system given in the input is valid, i.e., it satisfies conditions (1) and (2) from the easy version.

The first line of the input contains two space-separated integers n and m (1 ≤ n, m ≤ 100000) denoting the number of brains (which are conveniently numbered from 1 to n) and the number of brain connectors in the nervous system, respectively. In the next m lines, descriptions of brain connectors follow. Every connector is given as a pair of brains a b it connects (1 ≤ a, b ≤ n and a ≠ b).


Breaking news from zombie neurology! It turns out that – contrary to previous beliefs – every zombie is born with a single brain, and only later it evolves into a complicated brain structure. In fact, whenever a zombie consumes a brain, a new brain appears in its nervous system and gets immediately connected to one of the already existing brains using a single brain connector. Researchers are now interested in monitoring the brain latency of a zombie. Your task is to write a program which, given a history of evolution of a zombie's nervous system, computes its brain latency at every stage.

The first line of the input contains one number n – the number of brains in the final nervous system (2 ≤ n ≤ 200000). In the second line a history of zombie's nervous system evolution is given. For convenience, we number all the brains by 1, 2, ..., n in the same order as they appear in the nervous system (the zombie is born with a single brain, number 1, and subsequently brains 2, 3, ..., n are added). The second line contains n - 1 space-separated numbers p2, p3, ..., pn, meaning that after a new brain k is added to the system, it gets connected to a parent-brain .

Output n - 1 space-separated numbers – the brain latencies after the brain number k is added, for k = 2, 3, ..., n.


Zombies seem to have become much more intelligent lately – a few have somehow wandered into the base through the automatic gate. Heidi has had to beef up security, and a new gate has been installed. Unfortunately, now the questions being asked are more complicated, and even humans have trouble answering them. Can you still program the robot army to do this reliably?The new questions are of the following form: a grayscale photograph has been divided into several horizontal pieces, which have been arbitrarily rearranged. The task is to assemble the original image back from these pieces (somewhat like in a jigsaw puzzle). To further delay the zombies, significant Gaussian-distributed noise has been added to the image.

The input format is the same as in the previous version, except that the first line of every question now contains three space-separated numbers h, w and k (1 ≤ h, w ≤ 600, 2 ≤ k ≤ 16) – the height (number of rows) and width (number of columns) of the photograph and the number of pieces, respectively. The number of pieces evenly divides the height, and each piece is of the same height h / k. Again, there is only one input file to be processed, and the same resources are provided to you as in the previous version (except that now you are given all input images in .bmp format, rather than the first 50).

Your program should print q lines. The i-th line should contain your answer for the i-th question: a space-separated sequence of k numbers π1, π2, ..., πk such that:    π is a permutation of {1, 2, ..., k}, that is, each number from 1 to k appears exactly once in π,  for each j = 1, ..., k, πj is the position (index), in the original image, of the piece which is at position j in the input image. (See the illustration below for clarity.)   The second image from the test set. If the three pieces in the original image are numbered 1, 2, 3 from top to bottom, then the numbering in the image on the right should be 2, 3, 1. The correct answer for this image is thus 2 3 1.  Again, your answers will be accepted if they conform to this format and if at least 75% of them are correct. Again, you may process the input locally and submit just your precomputed answers (i.e., a program which just prints your output for the input file all.in).

NoteThe link to download all the necessary materials is http://assets.codeforces.com/files/690/medium_contestant_package.zip


So many wall designs to choose from! Even modulo 106 + 3, it's an enormous number. Given that recently Heidi acquired an unlimited supply of bricks, her choices are endless! She really needs to do something to narrow them down.Heidi is quick to come up with criteria for a useful wall:  In a useful wall, at least one segment is wider than W bricks. This should give the zombies something to hit their heads against. Or,  in a useful wall, at least one column is higher than H bricks. This provides a lookout from which zombies can be spotted at a distance. This should rule out a fair amount of possibilities, right? Help Heidi compute the number of useless walls that do not confirm to either of these criteria. In other words, a wall is useless if every segment has width at most W and height at most H.Parameter C, the total width of the wall, has the same meaning as in the easy version. However, note that the number of bricks is now unlimited.Output the number of useless walls modulo 106 + 3.

The first and the only line of the input contains three space-separated integers C, W and H (1 ≤ C ≤ 108, 1 ≤ W, H ≤ 100).

Output the number of different walls, modulo 106 + 3, which are useless according to Heidi's criteria.

NoteIf there is no brick in any of the columns, the structure is considered as a useless wall.


Heidi the Cow is aghast: cracks in the northern Wall? Zombies gathering outside, forming groups, preparing their assault? This must not happen! Quickly, she fetches her HC2 (Handbook of Crazy Constructions) and looks for the right chapter:How to build a wall:  Take a set of bricks.  Select one of the possible wall designs. Computing the number of possible designs is left as an exercise to the reader.  Place bricks on top of each other, according to the chosen design. This seems easy enough. But Heidi is a Coding Cow, not a Constructing Cow. Her mind keeps coming back to point 2b. Despite the imminent danger of a zombie onslaught, she wonders just how many possible walls she could build with up to n bricks.A wall is a set of wall segments as defined in the easy version. How many different walls can be constructed such that the wall consists of at least 1 and at most n bricks? Two walls are different if there exist a column c and a row r such that one wall has a brick in this spot, and the other does not.Along with n, you will be given C, the width of the wall (as defined in the easy version). Return the number of different walls modulo 106 + 3.

The first line contains two space-separated integers n and C, 1 ≤ n ≤ 500000, 1 ≤ C ≤ 200000.

Print the number of different walls that Heidi could build, modulo 106 + 3.

NoteThe number 106 + 3 is prime.In the second sample case, the five walls are:             B        BB., .B, BB, B., and .BIn the third sample case, the nine walls are the five as in the second sample case and in addition the following four: B    BB    B  B        BB., .B, BB, and BB


The Human-Cow Confederation (HC2), led by Heidi, has built a base where people and cows can hide, guarded from zombie attacks. The entrance to the base is protected by an automated gate which performs a kind of a Turing test: it shows the entering creature a photograph and asks them whether the top and bottom halves of this photograph have been swapped or not. A person (or even a cow) will have no problem answering such questions; on the other hand, a zombie would just randomly smash one of the two buttons.The creature is asked a series of such questions. If at least 75% of them are answered correctly, the gate is unlocked; otherwise, a side door opens, beneath which a huge fan is spinning...Heidi is now building a robot army to fight the zombies, and she wants the robots to also be able to enter the base. You are tasked with programming them to distinguish the images. The first two images from the test set. The first picture has been rearranged, but not the second.

The first line of the input contains the number q of questions (1 ≤ q ≤ 220). After that, q questions follow, each of which in the format described below. The first line of every question contains two space-separated integers h and w (1 ≤ h, w ≤ 600) – the height (number of rows) and width (number of columns) of the photograph. (Most photographs are roughly 200 × 300.) After this, h lines follow, each describing a single row of the picture. The picture is monochrome (in shades of grey). Its i-th row is described by w space-separated integers aij (j = 1, ..., w), where aij is the brightness of the corresponding pixel (0 ≤ aij < 256, where 0 is black and 255 is white). Each picture will be either a real-life photograph, or a real-life photograph which has been broken up into two pieces and rearranged. More precisely, in the latter case, the topmost  rows have been moved to the bottom of the picture. It is guaranteed that h is even. There is only a single input file to be processed, called all.in, and it is downloadable from the online judge. You are also a given another input file, called sample.in, which contains the first 20 pictures from all.in; you are provided the correct answers for sample.in in sample.out. You are also given a directory easy_bmp, which contains the first 50 input photographs in the form of .bmp image files, as well as a directory easy_sample_original_bmp, which contains the first 20 images before rearrangement. Check the notes for the download links.

Your program should print q lines. The i-th line should contain your answer for the i-th question: YES if the photograph has been rearranged and NO otherwise. Your answers will be accepted if they all conform to this format and if at least 75% of them are correct. Because the input is rather huge, feel free to process it locally and submit just your precomputed answers (i.e., a program which just prints your output for the input file all.in).

NoteThe link to download all necessary files is http://assets.codeforces.com/files/690/easy_contestant_package.zip


Heidi has finally found the mythical Tree of Life – a legendary combinatorial structure which is said to contain a prophecy crucially needed to defeat the undead armies.On the surface, the Tree of Life is just a regular undirected tree well-known from computer science. This means that it is a collection of n points (called vertices), some of which are connected using n - 1 line segments (edges) so that each pair of vertices is connected by a path (a sequence of one or more edges).To decipher the prophecy, Heidi needs to perform a number of steps. The first is counting the number of lifelines in the tree – these are paths of length 2, i.e., consisting of two edges. Help her!

The first line of the input contains a single integer n – the number of vertices in the tree (1 ≤ n ≤ 10000). The vertices are labeled with the numbers from 1 to n. Then n - 1 lines follow, each describing one edge using two space-separated numbers a b – the labels of the vertices connected by the edge (1 ≤ a < b ≤ n). It is guaranteed that the input represents a tree.

Print one integer – the number of lifelines in the tree.

NoteIn the second sample, there are four lifelines: paths between vertices 1 and 3, 2 and 4, 2 and 5, and 4 and 5.


According to rules of the Berland fashion, a jacket should be fastened by all the buttons except only one, but not necessarily it should be the last one. Also if the jacket has only one button, it should be fastened, so the jacket will not swinging open.You are given a jacket with n buttons. Determine if it is fastened in a right way.

The first line contains integer n (1 ≤ n ≤ 1000) — the number of buttons on the jacket. The second line contains n integers ai (0 ≤ ai ≤ 1). The number ai = 0 if the i-th button is not fastened. Otherwise ai = 1.

In the only line print the word "YES" if the jacket is fastened in a right way. Otherwise print the word "NO".


Let's call a string "s-palindrome" if it is symmetric about the middle of the string. For example, the string "oHo" is "s-palindrome", but the string "aa" is not. The string "aa" is not "s-palindrome", because the second half of it is not a mirror reflection of the first half.  English alphabet You are given a string s. Check if the string is "s-palindrome".

The only line contains the string s (1 ≤ |s| ≤ 1000) which consists of only English letters.

Print "TAK" if the string s is "s-palindrome" and "NIE" otherwise.


Heidi got tired of deciphering the prophecy hidden in the Tree of Life and decided to go back to her headquarters, rest a little and try there. Of course, she cannot uproot the Tree and take it with her, so she made a drawing of the Tree on a piece of paper. On second thought, she made more identical drawings so as to have n in total (where n is the number of vertices of the Tree of Life) – who knows what might happen?Indeed, on her way back Heidi was ambushed by a group of zombies. While she managed to fend them off, they have damaged her drawings in a peculiar way: from the i-th copy, the vertex numbered i was removed, along with all adjacent edges. In each picture, the zombies have also erased all the vertex numbers and relabeled the remaining n - 1 vertices arbitrarily using numbers 1 to n (fortunately, each vertex still has a distinct number). What's more, the drawings have been arbitrarily shuffled/reordered.Now Heidi wants to recover the Tree of Life from her descriptions of all the drawings (as lists of edges).

The first line of the input contains Z ≤ 20 – the number of test cases. Z descriptions of single test cases follow. In each test case, the first line of input contains numbers n (2 ≤ n ≤ 100) and k (where k is the number of drawings; we have k = n). In the following lines, the descriptions of the k drawings are given. The description of the i-th drawing is a line containing mi – the number of edges in this drawing, followed by mi lines describing edges, each of which contains two space-separated integers –- the numbers of the two vertices connected by the edge.

If Heidi's drawings cannot possibly come from a single tree, you should output the word NO. Otherwise, output one line containing the word YES and n - 1 lines describing any tree that Heidi's drawings could have come from. For every edge you should output the numbers of the vertices that it connects, separated with a single space. If there are many solutions, print any of them.


You are given a positive decimal number x.Your task is to convert it to the "simple exponential notation".Let x = a·10b, where 1 ≤ a < 10, then in general case the "simple exponential notation" looks like "aEb". If b equals to zero, the part "Eb" should be skipped. If a is an integer, it should be written without decimal point. Also there should not be extra zeroes in a and b.

The only line contains the positive decimal number x. The length of the line will not exceed 106. Note that you are given too large number, so you can't use standard built-in data types "float", "double" and other.

Print the only line — the "simple exponential notation" of the given number x.


You are given n integers a1,  a2,  ...,  an.A sequence of integers x1,  x2,  ...,  xk is called a "xor-sequence" if for every 1  ≤  i  ≤  k - 1 the number of ones in the binary representation of the number xi  xi  +  1's is a multiple of 3 and  for all 1 ≤ i ≤ k. The symbol  is used for the binary exclusive or operation.How many "xor-sequences" of length k exist? Output the answer modulo 109 + 7.Note if a = [1, 1] and k = 1 then the answer is 2, because you should consider the ones from a as different.

The first line contains two integers n and k (1 ≤ n ≤ 100, 1 ≤ k ≤ 1018) — the number of given integers and the length of the "xor-sequences". The second line contains n integers ai (0 ≤ ai ≤ 1018).

Print the only integer c — the number of "xor-sequences" of length k modulo 109 + 7.


Couple Cover, a wildly popular luck-based game, is about to begin! Two players must work together to construct a rectangle. A bag with n balls, each with an integer written on it, is placed on the table. The first player reaches in and grabs a ball randomly (all balls have equal probability of being chosen) — the number written on this ball is the rectangle's width in meters. This ball is not returned to the bag, and the second player reaches into the bag and grabs another ball — the number written on this ball is the rectangle's height in meters. If the area of the rectangle is greater than or equal some threshold p square meters, the players win. Otherwise, they lose.The organizers of the game are trying to select an appropriate value for p so that the probability of a couple winning is not too high and not too low, but they are slow at counting, so they have hired you to answer some questions for them. You are given a list of the numbers written on the balls, the organizers would like to know how many winning pairs of balls exist for different values of p. Note that two pairs are different if either the first or the second ball is different between the two in pair, and two different balls with the same number are considered different.

The input begins with a single positive integer n in its own line (1 ≤ n ≤ 106). The second line contains n positive integers — the i-th number in this line is equal to ai (1 ≤ ai ≤ 3·106), the number written on the i-th ball. The next line contains an integer m (1 ≤ m ≤ 106), the number of questions you are being asked. Then, the following line contains m positive integers — the j-th number in this line is equal to the value of p (1 ≤ p ≤ 3·106) in the j-th question you are being asked.

For each question, print the number of winning pairs of balls that exist for the given value of p in the separate line.


You are given a permutation of the numbers 1, 2, ..., n and m pairs of positions (aj, bj).At each step you can choose a pair from the given positions and swap the numbers in that positions. What is the lexicographically maximal permutation one can get?Let p and q be two permutations of the numbers 1, 2, ..., n. p is lexicographically smaller than the q if a number 1 ≤ i ≤ n exists, so pk = qk for 1 ≤ k < i and pi < qi.

The first line contains two integers n and m (1 ≤ n, m ≤ 106) — the length of the permutation p and the number of pairs of positions. The second line contains n distinct integers pi (1 ≤ pi ≤ n) — the elements of the permutation p. Each of the last m lines contains two integers (aj, bj) (1 ≤ aj, bj ≤ n) — the pairs of positions to swap. Note that you are given a positions, not the values to swap.

Print the only line with n distinct integers p'i (1 ≤ p'i ≤ n) — the lexicographically maximal permutation one can get.


To add insult to injury, the zombies have taken all but two drawings from Heidi! Please help her recover the Tree of Life from only these two drawings.

The input format is the same as in the medium version, except that now the bound on n is 2 ≤ n ≤ 1000 and that k = 2.


Barney was hanging out with Nora for a while and now he thinks he may have feelings for her. Barney wants to send her a cheesy text message and wants to make her as happy as possible.  Initially, happiness level of Nora is 0. Nora loves some pickup lines like "I'm falling for you" and stuff. Totally, she knows n pickup lines, each consisting only of lowercase English letters, also some of them may be equal (in writing, but different in pronouncing or meaning though). Every time Nora sees i-th pickup line as a consecutive subsequence of Barney's text message her happiness level increases by ai. These substrings may overlap, for example, Nora will see the pickup line aa twice and the pickup line ab once in text message aaab.Due to texting app limits, Barney's text may have up to l characters.Barney asked you to help him make Nora as much happy as possible, it's gonna be legen...

The first line of input contains two integers n and l (1 ≤ n ≤ 200, 1 ≤ l ≤ 1014) — the number of pickup lines and the maximum length of Barney's text. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 100), meaning that Nora's happiness level increases by ai after every time seeing i-th pickup line. The next n lines contain the pickup lines. i-th of them contains a single string si consisting of only English lowercase letter. Summary length of all pickup lines does not exceed 200. All strings are not empty.

Print the only integer — the maximum possible value of Nora's happiness level after reading Barney's text.

NoteAn optimal answer for the first sample case is hearth containing each pickup line exactly once.An optimal answer for the second sample case is artart.


Barney is searching for his dream girl. He lives in NYC. NYC has n junctions numbered from 1 to n and n - 1 roads connecting them. We will consider the NYC as a rooted tree with root being junction 1. m girls live in NYC, i-th of them lives along junction ci and her weight initially equals i pounds.  Barney consider a girl x to be better than a girl y if and only if: girl x has weight strictly less than girl y or girl x and girl y have equal weights and index of girl x living junction index is strictly less than girl y living junction index, i.e. cx < cy. Thus for any two girls one of them is always better than another one.For the next q days, one event happens each day. There are two types of events:  Barney goes from junction v to junction u. As a result he picks at most k best girls he still have not invited from junctions on his way and invites them to his house to test if one of them is his dream girl. If there are less than k not invited girls on his path, he invites all of them. Girls living along junctions in subtree of junction v (including v itself) put on some weight. As result, their weights increase by k pounds. Your task is for each event of first type tell Barney the indices of girls he will invite to his home in this event.

The first line of input contains three integers n, m and q (1 ≤ n, m, q ≤ 105) — the number of junctions in NYC, the number of girls living in NYC and the number of events respectively. The next n - 1 lines describes the roads. Each line contains two integers v and u (1 ≤ v, u ≤ n, v ≠ u) meaning that there is a road connecting junctions v and u . The next line contains m integers c1, c2, ..., cm (1 ≤ ci ≤ n) — the girl's living junctions. The next q lines describe the events in chronological order. Each line starts with an integer t (1 ≤ t ≤ 2) — type of the event . If t = 1 then the line describes event of first type three integers v, u and k (1 ≤ v, u, k ≤ n) follow — the endpoints of Barney's path and the number of girls that he will invite at most. Otherwise the line describes event of second type and two integers v and k (1 ≤ v ≤ n, 1 ≤ k ≤ 109) follow — the root of the subtree and value by which all the girls' weights in the subtree should increase.

For each event of the first type, print number t and then t integers g1, g2, ..., gt in one line, meaning that in this event Barney will invite t girls whose indices are g1, ..., gt in the order from the best to the worst according to Barney's considerations.

NoteFor the first sample case:  Description of events:  Weights of girls in subtree of junction 4 increase by 3. These girls have IDs: 1, 3, 5, 4, 7.  Barney goes from junction 2 to 1. Girls on his way have IDs 1, 2, 3, 5, 6, 7 with weights 4, 2, 6, 8, 6, 10 respectively. So, he invites girls 2 and 1.  Barney goes from junction 4 to junction 2. Girls on his way has IDs 3, 5, 7 with weights 6, 8, 10 respectively. So he invites girl 3.  Weight of girls in subtree of junction 2 increase by 10. There are no not invited girls, so nothing happens.  Weight of girls in subtree of junction 1 increase by 10. These girls (all girls left) have IDs: 4, 5, 6, 7.  Barney goes from junction 2 to junction 4. Girls on his way has IDs 5, 7 with weights 18, 20 respectively. So he invites girl 5.  Barney goes from junction 2 to junction 3. There is no girl on his way.  Weight of girls in subtree of junction 5 increase by 2. The only girl there is girl with ID 4.  Weight of girls in subtree of junction 4 increase by 9. These girls have IDs: 4, 6, 7.  Barney goes from junction 3 to junction 5. Only girl on his way is girl with ID 4.  Barney goes from junction 1 to junction 2. Girls on his way has IDs 6, 7 with weights 16, 29 respectively.


Barney lives in NYC. NYC has infinite number of intersections numbered with positive integers starting from 1. There exists a bidirectional road between intersections i and 2i and another road between i and 2i + 1 for every positive integer i. You can clearly see that there exists a unique shortest path between any two intersections.  Initially anyone can pass any road for free. But since SlapsGiving is ahead of us, there will q consecutive events happen soon. There are two types of events:1. Government makes a new rule. A rule can be denoted by integers v, u and w. As the result of this action, the passing fee of all roads on the shortest path from u to v increases by w dollars. 2. Barney starts moving from some intersection v and goes to intersection u where there's a girl he wants to cuddle (using his fake name Lorenzo Von Matterhorn). He always uses the shortest path (visiting minimum number of intersections or roads) between two intersections.Government needs your calculations. For each time Barney goes to cuddle a girl, you need to tell the government how much money he should pay (sum of passing fee of all roads he passes).

The first line of input contains a single integer q (1 ≤ q ≤ 1 000). The next q lines contain the information about the events in chronological order. Each event is described in form 1 v u w if it's an event when government makes a new rule about increasing the passing fee of all roads on the shortest path from u to v by w dollars, or in form 2 v u if it's an event when Barnie goes to cuddle from the intersection v to the intersection u. 1 ≤ v, u ≤ 1018, v ≠ u, 1 ≤ w ≤ 109 states for every description line.

For each event of second type print the sum of passing fee of all roads Barney passes in this event, in one line. Print the answers in chronological order of corresponding events.

NoteIn the example testcase:Here are the intersections used:    Intersections on the path are 3, 1, 2 and 4.  Intersections on the path are 4, 2 and 1.  Intersections on the path are only 3 and 6.  Intersections on the path are 4, 2, 1 and 3. Passing fee of roads on the path are 32, 32 and 30 in order. So answer equals to 32 + 32 + 30 = 94.  Intersections on the path are 6, 3 and 1.  Intersections on the path are 3 and 7. Passing fee of the road between them is 0.  Intersections on the path are 2 and 4. Passing fee of the road between them is 32 (increased by 30 in the first event and by 2 in the second).


As we all know Barney's job is "PLEASE" and he has not much to do at work. That's why he started playing "cups and key". In this game there are three identical cups arranged in a line from left to right. Initially key to Barney's heart is under the middle cup.  Then at one turn Barney swaps the cup in the middle with any of other two cups randomly (he choses each with equal probability), so the chosen cup becomes the middle one. Game lasts n turns and Barney independently choses a cup to swap with the middle one within each turn, and the key always remains in the cup it was at the start.After n-th turn Barney asks a girl to guess which cup contains the key. The girl points to the middle one but Barney was distracted while making turns and doesn't know if the key is under the middle cup. That's why he asked you to tell him the probability that girl guessed right.Number n of game turns can be extremely large, that's why Barney did not give it to you. Instead he gave you an array a1, a2, ..., ak such that   in other words, n is multiplication of all elements of the given array.Because of precision difficulties, Barney asked you to tell him the answer as an irreducible fraction. In other words you need to find it as a fraction p / q such that , where  is the greatest common divisor. Since p and q can be extremely large, you only need to find the remainders of dividing each of them by 109 + 7.Please note that we want  of p and q to be 1, not  of their remainders after dividing by 109 + 7.

The first line of input contains a single integer k (1 ≤ k ≤ 105) — the number of elements in array Barney gave you. The second line contains k integers a1, a2, ..., ak (1 ≤ ai ≤ 1018) — the elements of the array.

In the only line of output print a single string x / y where x is the remainder of dividing p by 109 + 7 and y is the remainder of dividing q by 109 + 7.


Barney lives in country USC (United States of Charzeh). USC has n cities numbered from 1 through n and n - 1 roads between them. Cities and roads of USC form a rooted tree (Barney's not sure why it is rooted). Root of the tree is the city number 1. Thus if one will start his journey from city 1, he can visit any city he wants by following roads.  Some girl has stolen Barney's heart, and Barney wants to find her. He starts looking for in the root of the tree and (since he is Barney Stinson not a random guy), he uses a random DFS to search in the cities. A pseudo code of this algorithm is as follows:let starting_time be an array of length ncurrent_time = 0dfs(v):	current_time = current_time + 1	starting_time[v] = current_time	shuffle children[v] randomly (each permutation with equal possibility)	// children[v] is vector of children cities of city v	for u in children[v]:		dfs(u)As told before, Barney will start his journey in the root of the tree (equivalent to call dfs(1)).Now Barney needs to pack a backpack and so he wants to know more about his upcoming journey: for every city i, Barney wants to know the expected value of starting_time[i]. He's a friend of Jon Snow and knows nothing, that's why he asked for your help.

The first line of input contains a single integer n (1 ≤ n ≤ 105) — the number of cities in USC. The second line contains n - 1 integers p2, p3, ..., pn (1 ≤ pi < i), where pi is the number of the parent city of city number i in the tree, meaning there is a road between cities numbered pi and i in USC.

In the first and only line of output print n numbers, where i-th number is the expected value of starting_time[i]. Your answer for each city will be considered correct if its absolute or relative error does not exceed 10 - 6.


Ted has a pineapple. This pineapple is able to bark like a bulldog! At time t (in seconds) it barks for the first time. Then every s seconds after it, it barks twice with 1 second interval. Thus it barks at times t, t + s, t + s + 1, t + 2s, t + 2s + 1, etc.  Barney woke up in the morning and wants to eat the pineapple, but he can't eat it when it's barking. Barney plans to eat it at time x (in seconds), so he asked you to tell him if it's gonna bark at that time.

The first and only line of input contains three integers t, s and x (0 ≤ t, x ≤ 109, 2 ≤ s ≤ 109) — the time the pineapple barks for the first time, the pineapple barking interval, and the time Barney wants to eat the pineapple respectively.

Print a single "YES" (without quotes) if the pineapple will bark at time x or a single "NO" (without quotes) otherwise in the only line of output.

NoteIn the first and the second sample cases pineapple will bark at moments 3, 13, 14, ..., so it won't bark at the moment 4 and will bark at the moment 3.In the third and fourth sample cases pineapple will bark at moments 3, 11, 12, 19, 20, 27, 28, 35, 36, 43, 44, 51, 52, 59, ..., so it will bark at both moments 51 and 52.


Barney is standing in a bar and starring at a pretty girl. He wants to shoot her with his heart arrow but he needs to know the distance between him and the girl to make his shot accurate.  Barney asked the bar tender Carl about this distance value, but Carl was so busy talking to the customers so he wrote the distance value (it's a real number) on a napkin. The problem is that he wrote it in scientific notation. The scientific notation of some real number x is the notation of form AeB, where A is a real number and B is an integer and x = A × 10B is true. In our case A is between 0 and 9 and B is non-negative.Barney doesn't know anything about scientific notation (as well as anything scientific at all). So he asked you to tell him the distance value in usual decimal representation with minimal number of digits after the decimal point (and no decimal point if it is an integer). See the output format for better understanding.

The first and only line of input contains a single string of form a.deb where a, d and b are integers and e is usual character 'e' (0 ≤ a ≤ 9, 0 ≤ d < 10100, 0 ≤ b ≤ 100) — the scientific notation of the desired distance value. a and b contain no leading zeros and d contains no trailing zeros (but may be equal to 0). Also, b can not be non-zero if a is zero.

Print the only real number x (the desired distance value) in the only line in its decimal notation.  Thus if x is an integer, print it's integer value without decimal part and decimal point and without leading zeroes.  Otherwise print x in a form of p.q such that p is an integer that have no leading zeroes (but may be equal to zero), and q is an integer that have no trailing zeroes (and may not be equal to zero).


While creating high loaded systems one should pay a special attention to caching. This problem will be about one of the most popular caching algorithms called LRU (Least Recently Used).Suppose the cache may store no more than k objects. At the beginning of the workflow the cache is empty. When some object is queried we check if it is present in the cache and move it here if it's not. If there are more than k objects in the cache after this, the least recently used one should be removed. In other words, we remove the object that has the smallest time of the last query.Consider there are n videos being stored on the server, all of the same size. Cache can store no more than k videos and caching algorithm described above is applied. We know that any time a user enters the server he pick the video i with probability pi. The choice of the video is independent to any events before.The goal of this problem is to count for each of the videos the probability it will be present in the cache after 10100 queries.

The first line of the input contains two integers n and k (1 ≤ k ≤ n ≤ 20) — the number of videos and the size of the cache respectively. Next line contains n real numbers pi (0 ≤ pi ≤ 1), each of them is given with no more than two digits after decimal point. It's guaranteed that the sum of all pi is equal to 1.

Print n real numbers, the i-th of them should be equal to the probability that the i-th video will be present in the cache after 10100 queries. You answer will be considered correct if its absolute or relative error does not exceed 10 - 6.  Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .


A tree is an undirected connected graph without cycles.Let's consider a rooted undirected tree with n vertices, numbered 1 through n. There are many ways to represent such a tree. One way is to create an array with n integers p1, p2, ..., pn, where pi denotes a parent of vertex i (here, for convenience a root is considered its own parent).    For this rooted tree the array p is [2, 3, 3, 2]. Given a sequence p1, p2, ..., pn, one is able to restore a tree:  There must be exactly one index r that pr = r. A vertex r is a root of the tree.  For all other n - 1 vertices i, there is an edge between vertex i and vertex pi. A sequence p1, p2, ..., pn is called valid if the described procedure generates some (any) rooted tree. For example, for n = 3 sequences (1,2,2), (2,3,1) and (2,1,3) are not valid.You are given a sequence a1, a2, ..., an, not necessarily valid. Your task is to change the minimum number of elements, in order to get a valid sequence. Print the minimum number of changes and an example of a valid sequence after that number of changes. If there are many valid sequences achievable in the minimum number of changes, print any of them.

The first line of the input contains an integer n (2 ≤ n ≤ 200 000) — the number of vertices in the tree. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ n).

In the first line print the minimum number of elements to change, in order to get a valid sequence. In the second line, print any valid sequence possible to get from (a1, a2, ..., an) in the minimum number of changes. If there are many such sequences, any of them will be accepted.

NoteIn the first sample, it's enough to change one element. In the provided output, a sequence represents a tree rooted in a vertex 4 (because p4 = 4), which you can see on the left drawing below. One of other correct solutions would be a sequence 2 3 3 2, representing a tree rooted in vertex 3 (right drawing below). On both drawings, roots are painted red.  In the second sample, the given sequence is already valid.


Barney has finally found the one, a beautiful young lady named Lyanna. The problem is, Lyanna and Barney are trapped in Lord Loss' castle. This castle has shape of a convex polygon of n points. Like most of castles in Demonata worlds, this castle has no ceiling.  Barney and Lyanna have an escape plan, but it requires some geometry knowledge, so they asked for your help.Barney knows that demons are organized and move in lines. He and Lyanna want to wait for the appropriate time so they need to watch for the demons. Each of them wants to stay in a point inside the castle (possibly on edges or corners), also they may stay in the same position. They both want to pick a real number r and watch all points in the circles with radius r around each of them (these two circles may overlap).  We say that Barney and Lyanna are watching carefully if and only if for every edge of the polygon, at least one of them can see at least one point on the line this edge lies on, thus such point may not be on the edge but it should be on edge's line. Formally, each edge line should have at least one common point with at least one of two circles.The greater r is, the more energy and focus they need. So they asked you to tell them the minimum value of r such that they can watch carefully.

The first line of input contains a single integer n (3 ≤ n ≤ 300) — the number of castle polygon vertices. The next n lines describe the polygon vertices in counter-clockwise order. i-th of them contains two integers xi and yi (|xi|, |yi| ≤ 104) — the coordinates of i-th point of the castle. It is guaranteed that given points form a convex polygon, in particular, any three of them do not line on the same line.

In the first line print the single number r — minimum radius of guys' watching circles. In the second line print the pair of coordinates of point where Barney should stay. In the third line print the pair of coordinates of point where Lyanna should stay. Points should lie inside the polygon. Coordinates may not be integers. If there are multiple answers print any of them. Your answer will be considered correct if its absolute or relative error doesn't exceed 10 - 6.

NoteIn the first example guys can stay in opposite corners of the castle.


Bearland is a dangerous place. Limak can’t travel on foot. Instead, he has k magic teleportation stones. Each stone can be used at most once. The i-th stone allows to teleport to a point (axi, ayi). Limak can use stones in any order.There are n monsters in Bearland. The i-th of them stands at (mxi, myi).The given k + n points are pairwise distinct.After each teleportation, Limak can shoot an arrow in some direction. An arrow will hit the first monster in the chosen direction. Then, both an arrow and a monster disappear. It’s dangerous to stay in one place for long, so Limak can shoot only one arrow from one place.A monster should be afraid if it’s possible that Limak will hit it. How many monsters should be afraid of Limak?

The first line of the input contains two integers k and n (1 ≤ k ≤ 7, 1 ≤ n ≤ 1000) — the number of stones and the number of monsters. The i-th of following k lines contains two integers axi and ayi ( - 109 ≤ axi, ayi ≤ 109) — coordinates to which Limak can teleport using the i-th stone. The i-th of last n lines contains two integers mxi and myi ( - 109 ≤ mxi, myi ≤ 109) — coordinates of the i-th monster. The given k + n points are pairwise distinct.

Print the number of monsters which should be afraid of Limak.

NoteIn the first sample, there are two stones and four monsters. Stones allow to teleport to points ( - 2,  - 1) and (4, 5), marked blue in the drawing below. Monsters are at (4, 2), (2, 1), (4,  - 1) and (1,  - 1), marked red. A monster at (4,  - 1) shouldn't be afraid because it's impossible that Limak will hit it with an arrow. Other three monsters can be hit and thus the answer is 3.  In the second sample, five monsters should be afraid. Safe monsters are those at (300, 600), (170, 340) and (90, 180).


Vasya has n days of vacations! So he decided to improve his IT skills and do sport. Vasya knows the following information about each of this n days: whether that gym opened and whether a contest was carried out in the Internet on that day. For the i-th day there are four options:  on this day the gym is closed and the contest is not carried out;  on this day the gym is closed and the contest is carried out;  on this day the gym is open and the contest is not carried out;  on this day the gym is open and the contest is carried out. On each of days Vasya can either have a rest or write the contest (if it is carried out on this day), or do sport (if the gym is open on this day).Find the minimum number of days on which Vasya will have a rest (it means, he will not do sport and write the contest at the same time). The only limitation that Vasya has — he does not want to do the same activity on two consecutive days: it means, he will not do sport on two consecutive days, and write the contest on two consecutive days.

The first line contains a positive integer n (1 ≤ n ≤ 100) — the number of days of Vasya's vacations. The second line contains the sequence of integers a1, a2, ..., an (0 ≤ ai ≤ 3) separated by space, where:    ai equals 0, if on the i-th day of vacations the gym is closed and the contest is not carried out;  ai equals 1, if on the i-th day of vacations the gym is closed, but the contest is carried out;  ai equals 2, if on the i-th day of vacations the gym is open and the contest is not carried out;  ai equals 3, if on the i-th day of vacations the gym is open and the contest is carried out.

Print the minimum possible number of days on which Vasya will have a rest. Remember that Vasya refuses:   to do sport on any two consecutive days,  to write the contest on any two consecutive days.

NoteIn the first test Vasya can write the contest on the day number 1 and do sport on the day number 3. Thus, he will have a rest for only 2 days.In the second test Vasya should write contests on days number 1, 3, 5 and 7, in other days do sport. Thus, he will not have a rest for a single day.In the third test Vasya can do sport either on a day number 1 or number 2. He can not do sport in two days, because it will be contrary to the his limitation. Thus, he will have a rest for only one day.


Sometime the classic solution are not powerful enough and we have to design our own. For the purpose of this problem you have to implement the part of the system of task scheduling.Each task should be executed at some particular moments of time. In our system you may set the exact value for the second, minute, hour, day of the week, day and month, when the task should be executed. Moreover, one can set a special value -1 that means any value of this parameter is valid.For example, if the parameter string is -1 59 23 -1 -1 -1, the problem will be executed every day at 23:59:00, 23:59:01, 23:59:02, ..., 23:59:59 (60 times in total).Seconds, minutes and hours are numbered starting from zero, while day, months and days of the week are numbered starting from one. The first day of the week is Monday.There is one special case that is treated separately. If both day of the week and day are given (i.e. differ from -1) to execute the task only one of these two (at least one, if both match this is fine too) parameters should match the current time (of course, all other parameters should match too). For example, the string of parameters 0 0 12 6 3 7 means that the task will be executed both on Saturday, July 2nd, 2016 and on Sunday, July 3rd, 2016 at noon.One should not forget about the existence of the leap years. The year is leap if it's number is divisible by 400, or is not divisible by 100, but is divisible by 4. Each leap year has 366 days instead of usual 365, by extending February to 29 days rather than the common 28.The current time is represented as the number of seconds passed after 00:00:00 January 1st, 1970 (Thursday).You are given the string of six parameters, describing the moments of time the task should be executed. You are also given a number of moments of time. For each of them you have to find the first moment of time strictly greater than the current when the task will be executed.

The first line of the input contains six integers s, m, h, day, date and month (0 ≤ s, m ≤ 59, 0 ≤ h ≤ 23, 1 ≤ day ≤ 7, 1 ≤ date ≤ 31, 1 ≤ month ≤ 12). Each of the number can also be equal to  - 1. It's guaranteed, that there are infinitely many moments of time when this task should be executed. Next line contains the only integer n (1 ≤ n ≤ 1000) — the number of moments of time you have to solve the problem for. Each of the next n lines contains a single integer ti (0 ≤ ti ≤ 1012).

Print n lines, the i-th of them should contain the first moment of time strictly greater than ti, when the task should be executed.

NoteThe moment of time 1467372658 after the midnight of January 1st, 1970 is 11:30:58 July 1st, 2016.


Two positive integers are coprime if and only if they don't have a common divisor greater than 1.Some bear doesn't want to tell Radewoosh how to solve some algorithmic problem. So, Radewoosh is going to break into that bear's safe with solutions. To pass through the door, he must enter a permutation of numbers 1 through n. The door opens if and only if an entered permutation p1, p2, ..., pn satisfies:In other words, two different elements are coprime if and only if their indices are coprime. Some elements of a permutation may be already fixed. In how many ways can Radewoosh fill the remaining gaps so that the door will open? Print the answer modulo 109 + 7.

The first line of the input contains one integer n (2 ≤ n ≤ 1 000 000). The second line contains n integers p1, p2, ..., pn (0 ≤ pi ≤ n) where pi = 0 means a gap to fill, and pi ≥ 1 means a fixed number. It's guaranteed that if i ≠ j and pi, pj ≥ 1 then pi ≠ pj.

Print the number of ways to fill the gaps modulo 109 + 7 (i.e. modulo 1000000007).

NoteIn the first sample test, none of four element is fixed. There are four permutations satisfying the given conditions: (1,2,3,4), (1,4,3,2), (3,2,1,4), (3,4,1,2).In the second sample test, there must be p3 = 1 and p4 = 2. The two permutations satisfying the conditions are: (3,4,1,2,5), (5,4,1,2,3).


There will be a launch of a new, powerful and unusual collider very soon, which located along a straight line. n particles will be launched inside it. All of them are located in a straight line and there can not be two or more particles located in the same point. The coordinates of the particles coincide with the distance in meters from the center of the collider, xi is the coordinate of the i-th particle and its position in the collider at the same time. All coordinates of particle positions are even integers.You know the direction of each particle movement — it will move to the right or to the left after the collider's launch start. All particles begin to move simultaneously at the time of the collider's launch start. Each particle will move straight to the left or straight to the right with the constant speed of 1 meter per microsecond. The collider is big enough so particles can not leave it in the foreseeable time.Write the program which finds the moment of the first collision of any two particles of the collider. In other words, find the number of microseconds before the first moment when any two particles are at the same point.

The first line contains the positive integer n (1 ≤ n ≤ 200 000) — the number of particles.  The second line contains n symbols "L" and "R". If the i-th symbol equals "L", then the i-th particle will move to the left, otherwise the i-th symbol equals "R" and the i-th particle will move to the right. The third line contains the sequence of pairwise distinct even integers x1, x2, ..., xn (0 ≤ xi ≤ 109) — the coordinates of particles in the order from the left to the right. It is guaranteed that the coordinates of particles are given in the increasing order.

In the first line print the only integer — the first moment (in microseconds) when two particles are at the same point and there will be an explosion.  Print the only integer -1, if the collision of particles doesn't happen.

NoteIn the first sample case the first explosion will happen in 1 microsecond because the particles number 1 and 2 will simultaneously be at the same point with the coordinate 3. In the second sample case there will be no explosion because there are no particles which will simultaneously be at the same point.


On vacations n pupils decided to go on excursion and gather all together. They need to overcome the path with the length l meters. Each of the pupils will go with the speed equal to v1. To get to the excursion quickly, it was decided to rent a bus, which has seats for k people (it means that it can't fit more than k people at the same time) and the speed equal to v2. In order to avoid seasick, each of the pupils want to get into the bus no more than once.Determine the minimum time required for all n pupils to reach the place of excursion. Consider that the embarkation and disembarkation of passengers, as well as the reversal of the bus, take place immediately and this time can be neglected.

The first line of the input contains five positive integers n, l, v1, v2 and k (1 ≤ n ≤ 10 000, 1 ≤ l ≤ 109, 1 ≤ v1 < v2 ≤ 109, 1 ≤ k ≤ n) — the number of pupils, the distance from meeting to the place of excursion, the speed of each pupil, the speed of bus and the number of seats in the bus.

Print the real number — the minimum time in which all pupils can reach the place of excursion. Your answer will be considered correct if its absolute or relative error won't exceed 10 - 6.

NoteIn the first sample we should immediately put all five pupils to the bus. The speed of the bus equals 2 and the distance is equal to 10, so the pupils will reach the place of excursion in time 10 / 2 = 5.


Again, there are hard times in Berland! Many towns have such tensions that even civil war is possible. There are n towns in Reberland, some pairs of which connected by two-way roads. It is not guaranteed that it is possible to reach one town from any other town using these roads. Towns s and t announce the final break of any relationship and intend to rule out the possibility of moving between them by the roads. Now possibly it is needed to close several roads so that moving from s to t using roads becomes impossible. Each town agrees to spend money on closing no more than one road, therefore, the total number of closed roads will be no more than two.Help them find set of no more than two roads such that there will be no way between s and t after closing these roads. For each road the budget required for its closure was estimated. Among all sets find such that the total budget for the closure of a set of roads is minimum.

The first line of the input contains two integers n and m (2 ≤ n ≤ 1000, 0 ≤ m ≤ 30 000) — the number of towns in Berland and the number of roads. The second line contains integers s and t (1 ≤ s, t ≤ n, s ≠ t) — indices of towns which break up the relationships. Then follow m lines, each of them contains three integers xi, yi and wi (1 ≤ xi, yi ≤ n, 1 ≤ wi ≤ 109) — indices of towns connected by the i-th road, and the budget on its closure. All roads are bidirectional. It is allowed that the pair of towns is connected by more than one road. Roads that connect the city to itself are allowed.

In the first line print the minimum budget required to break up the relations between s and t, if it is allowed to close no more than two roads. In the second line print the value c (0 ≤ c ≤ 2) — the number of roads to be closed in the found solution. In the third line print in any order c diverse integers from 1 to m — indices of closed roads. Consider that the roads are numbered from 1 to m in the order they appear in the input.  If it is impossible to make towns s and t disconnected by removing no more than 2 roads, the output should contain a single line -1.  If there are several possible answers, you may print any of them.


You are given a description of a depot. It is a rectangular checkered field of n × m size. Each cell in a field can be empty (".") or it can be occupied by a wall ("*"). You have one bomb. If you lay the bomb at the cell (x, y), then after triggering it will wipe out all walls in the row x and all walls in the column y.You are to determine if it is possible to wipe out all walls in the depot by placing and triggering exactly one bomb. The bomb can be laid both in an empty cell or in a cell occupied by a wall.

The first line contains two positive integers n and m (1 ≤ n, m ≤ 1000) — the number of rows and columns in the depot field.  The next n lines contain m symbols "." and "*" each — the description of the field. j-th symbol in i-th of them stands for cell (i, j). If the symbol is equal to ".", then the corresponding cell is empty, otherwise it equals "*" and the corresponding cell is occupied by a wall.

If it is impossible to wipe out all walls by placing and triggering exactly one bomb, then print "NO" in the first line (without quotes). Otherwise print "YES" (without quotes) in the first line and two integers in the second line — the coordinates of the cell at which the bomb should be laid. If there are multiple answers, print any of them.


Treeland is a country in which there are n towns connected by n - 1 two-way road such that it's possible to get from any town to any other town. In Treeland there are 2k universities which are located in different towns. Recently, the president signed the decree to connect universities by high-speed network.The Ministry of Education understood the decree in its own way and decided that it was enough to connect each university with another one by using a cable. Formally, the decree will be done! To have the maximum sum in the budget, the Ministry decided to divide universities into pairs so that the total length of the required cable will be maximum. In other words, the total distance between universities in k pairs should be as large as possible. Help the Ministry to find the maximum total distance. Of course, each university should be present in only one pair. Consider that all roads have the same length which is equal to 1.

The first line of the input contains two integers n and k (2 ≤ n ≤ 200 000, 1 ≤ k ≤ n / 2) — the number of towns in Treeland and the number of university pairs. Consider that towns are numbered from 1 to n.  The second line contains 2k distinct integers u1, u2, ..., u2k (1 ≤ ui ≤ n) — indices of towns in which universities are located.  The next n - 1 line contains the description of roads. Each line contains the pair of integers xj and yj (1 ≤ xj, yj ≤ n), which means that the j-th road connects towns xj and yj. All of them are two-way roads. You can move from any town to any other using only these roads.

Print the maximum possible sum of distances in the division of universities into k pairs.

NoteThe figure below shows one of possible division into pairs in the first test. If you connect universities number 1 and 6 (marked in red) and universities number 2 and 5 (marked in blue) by using the cable, the total distance will equal 6 which will be the maximum sum in this example.


In an attempt to escape the Mischievous Mess Makers' antics, Farmer John has abandoned his farm and is traveling to the other side of Bovinia. During the journey, he and his k cows have decided to stay at the luxurious Grand Moo-dapest Hotel. The hotel consists of n rooms located in a row, some of which are occupied.Farmer John wants to book a set of k + 1 currently unoccupied rooms for him and his cows. He wants his cows to stay as safe as possible, so he wishes to minimize the maximum distance from his room to the room of his cow. The distance between rooms i and j is defined as |j - i|. Help Farmer John protect his cows by calculating this minimum possible distance.

The first line of the input contains two integers n and k (1 ≤ k < n ≤ 100 000) — the number of rooms in the hotel and the number of cows travelling with Farmer John. The second line contains a string of length n describing the rooms. The i-th character of the string will be '0' if the i-th room is free, and '1' if the i-th room is occupied. It is guaranteed that at least k + 1 characters of this string are '0', so there exists at least one possible choice of k + 1 rooms for Farmer John and his cows to stay in.

Print the minimum possible distance between Farmer John's room and his farthest cow.

NoteIn the first sample, Farmer John can book room 3 for himself, and rooms 1 and 4 for his cows. The distance to the farthest cow is 2. Note that it is impossible to make this distance 1, as there is no block of three consecutive unoccupied rooms.In the second sample, Farmer John can book room 1 for himself and room 3 for his single cow. The distance between him and his cow is 2.In the third sample, Farmer John books all three available rooms, taking the middle room for himself so that both cows are next to him. His distance from the farthest cow is 1.


While Farmer John rebuilds his farm in an unfamiliar portion of Bovinia, Bessie is out trying some alternative jobs. In her new gig as a reporter, Bessie needs to know about programming competition results as quickly as possible. When she covers the 2016 Robot Rap Battle Tournament, she notices that all of the robots operate under deterministic algorithms. In particular, robot i will beat robot j if and only if robot i has a higher skill level than robot j. And if robot i beats robot j and robot j beats robot k, then robot i will beat robot k. Since rapping is such a subtle art, two robots can never have the same skill level.Given the results of the rap battles in the order in which they were played, determine the minimum number of first rap battles that needed to take place before Bessie could order all of the robots by skill level.

The first line of the input consists of two integers, the number of robots n (2 ≤ n ≤ 100 000) and the number of rap battles m (). The next m lines describe the results of the rap battles in the order they took place. Each consists of two integers ui and vi (1 ≤ ui, vi ≤ n, ui ≠ vi), indicating that robot ui beat robot vi in the i-th rap battle. No two rap battles involve the same pair of robots. It is guaranteed that at least one ordering of the robots satisfies all m relations.

Print the minimum k such that the ordering of the robots by skill level is uniquely defined by the first k rap battles. If there exists more than one ordering that satisfies all m relations, output -1.

NoteIn the first sample, the robots from strongest to weakest must be (4, 2, 1, 3), which Bessie can deduce after knowing the results of the first four rap battles.In the second sample, both (1, 3, 2) and (3, 1, 2) are possible orderings of the robots from strongest to weakest after both rap battles.


After getting kicked out of her reporting job for not knowing the alphabet, Bessie has decided to attend school at the Fillet and Eggs Eater Academy. She has been making good progress with her studies and now knows the first k English letters.Each morning, Bessie travels to school along a sidewalk consisting of m + n tiles. In order to help Bessie review, Mr. Moozing has labeled each of the first m sidewalk tiles with one of the first k lowercase English letters, spelling out a string t. Mr. Moozing, impressed by Bessie's extensive knowledge of farm animals, plans to let her finish labeling the last n tiles of the sidewalk by herself.Consider the resulting string s (|s| = m + n) consisting of letters labeled on tiles in order from home to school. For any sequence of indices p1 < p2 < ... < pq we can define subsequence of the string s as string sp1sp2... spq. Two subsequences are considered to be distinct if they differ as strings. Bessie wants to label the remaining part of the sidewalk such that the number of distinct subsequences of tiles is maximum possible. However, since Bessie hasn't even finished learning the alphabet, she needs your help!Note that empty subsequence also counts.

The first line of the input contains two integers n and k (0 ≤ n ≤ 1 000 000, 1 ≤ k ≤ 26). The second line contains a string t (|t| = m, 1 ≤ m ≤ 1 000 000) consisting of only first k lowercase English letters.

Determine the maximum number of distinct subsequences Bessie can form after labeling the last n sidewalk tiles each with one of the first k lowercase English letters. Since this number can be rather large, you should print it modulo 109 + 7. Please note, that you are not asked to maximize the remainder modulo 109 + 7! The goal is to maximize the initial value and then print the remainder.

NoteIn the first sample, the optimal labeling gives 8 different subsequences: "" (the empty string), "a", "c", "b", "ac", "ab", "cb", and "acb".  In the second sample, the entire sidewalk is already labeled. The are 10 possible different subsequences: "" (the empty string), "a", "b", "aa", "ab", "ba", "aaa", "aab", "aba", and "aaba". Note that some strings, including "aa", can be obtained with multiple sequences of tiles, but are only counted once.


Watchmen are in a danger and Doctor Manhattan together with his friend Daniel Dreiberg should warn them as soon as possible. There are n watchmen on a plane, the i-th watchman is located at point (xi, yi).They need to arrange a plan, but there are some difficulties on their way. As you know, Doctor Manhattan considers the distance between watchmen i and j to be |xi - xj| + |yi - yj|. Daniel, as an ordinary person, calculates the distance using the formula .The success of the operation relies on the number of pairs (i, j) (1 ≤ i < j ≤ n), such that the distance between watchman i and watchmen j calculated by Doctor Manhattan is equal to the distance between them calculated by Daniel. You were asked to compute the number of such pairs.

The first line of the input contains the single integer n (1 ≤ n ≤ 200 000) — the number of watchmen. Each of the following n lines contains two integers xi and yi (|xi|, |yi| ≤ 109). Some positions may coincide.

Print the number of pairs of watchmen such that the distance between them calculated by Doctor Manhattan is equal to the distance calculated by Daniel.

NoteIn the first sample, the distance between watchman 1 and watchman 2 is equal to |1 - 7| + |1 - 5| = 10 for Doctor Manhattan and  for Daniel. For pairs (1, 1), (1, 5) and (7, 5), (1, 5) Doctor Manhattan and Daniel will calculate the same distances.


Vasya's telephone contains n photos. Photo number 1 is currently opened on the phone. It is allowed to move left and right to the adjacent photo by swiping finger over the screen. If you swipe left from the first photo, you reach photo n. Similarly, by swiping right from the last photo you reach photo 1. It takes a seconds to swipe from photo to adjacent.For each photo it is known which orientation is intended for it — horizontal or vertical. Phone is in the vertical orientation and can't be rotated. It takes b second to change orientation of the photo.Vasya has T seconds to watch photos. He want to watch as many photos as possible. If Vasya opens the photo for the first time, he spends 1 second to notice all details in it. If photo is in the wrong orientation, he spends b seconds on rotating it before watching it. If Vasya has already opened the photo, he just skips it (so he doesn't spend any time for watching it or for changing its orientation). It is not allowed to skip unseen photos.Help Vasya find the maximum number of photos he is able to watch during T seconds.

The first line of the input contains 4 integers n, a, b, T (1 ≤ n ≤ 5·105, 1 ≤ a, b ≤ 1000, 1 ≤ T ≤ 109) — the number of photos, time to move from a photo to adjacent, time to change orientation of a photo and time Vasya can spend for watching photo. Second line of the input contains a string of length n containing symbols 'w' and 'h'.  If the i-th position of a string contains 'w', then the photo i should be seen in the horizontal orientation. If the i-th position of a string contains 'h', then the photo i should be seen in vertical orientation.

Output the only integer, the maximum number of photos Vasya is able to watch during those T seconds.

NoteIn the first sample test you can rotate the first photo (3 seconds), watch the first photo (1 seconds), move left (2 second), rotate fourth photo (3 seconds), watch fourth photo (1 second). The whole process takes exactly 10 seconds.Note that in the last sample test the time is not enough even to watch the first photo, also you can't skip it.


Little Petya is now fond of data compression algorithms. He has already studied gz, bz, zip algorithms and many others. Inspired by the new knowledge, Petya is now developing the new compression algorithm which he wants to name dis.Petya decided to compress tables. He is given a table a consisting of n rows and m columns that is filled with positive integers. He wants to build the table a' consisting of positive integers such that the relative order of the elements in each row and each column remains the same. That is, if in some row i of the initial table ai, j < ai, k, then in the resulting table a'i, j < a'i, k, and if ai, j = ai, k then a'i, j = a'i, k. Similarly, if in some column j of the initial table ai, j < ap, j then in compressed table a'i, j < a'p, j and if ai, j = ap, j then a'i, j = a'p, j. Because large values require more space to store them, the maximum value in a' should be as small as possible.Petya is good in theory, however, he needs your help to implement the algorithm.

The first line of the input contains two integers n and m (, the number of rows and the number of columns of the table respectively. Each of the following n rows contain m integers ai, j (1 ≤ ai, j ≤ 109) that are the values in the table.

Output the compressed table in form of n lines each containing m integers. If there exist several answers such that the maximum number in the compressed table is minimum possible, you are allowed to output any of them.

NoteIn the first sample test, despite the fact a1, 2 ≠ a21, they are not located in the same row or column so they may become equal after the compression.


Vasya has decided to build a zip-line on trees of a nearby forest. He wants the line to be as long as possible but he doesn't remember exactly the heights of all trees in the forest. He is sure that he remembers correct heights of all trees except, possibly, one of them.It is known that the forest consists of n trees staying in a row numbered from left to right with integers from 1 to n. According to Vasya, the height of the i-th tree is equal to hi. The zip-line of length k should hang over k (1 ≤ k ≤ n) trees i1, i2, ..., ik (i1 < i2 < ... < ik) such that their heights form an increasing sequence, that is hi1 < hi2 < ... < hik.Petya had been in this forest together with Vasya, and he now has q assumptions about the mistake in Vasya's sequence h. His i-th assumption consists of two integers ai and bi indicating that, according to Petya, the height of the tree numbered ai is actually equal to bi. Note that Petya's assumptions are independent from each other.Your task is to find the maximum length of a zip-line that can be built over the trees under each of the q assumptions.In this problem the length of a zip line is considered equal to the number of trees that form this zip-line.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 400 000) — the number of the trees in the forest and the number of Petya's assumptions, respectively. The following line contains n integers hi (1 ≤ hi ≤ 109) — the heights of trees according to Vasya. Each of the following m lines contains two integers ai and bi (1 ≤ ai ≤ n, 1 ≤ bi ≤ 109).

For each of the Petya's assumptions output one integer, indicating the maximum length of a zip-line that can be built under this assumption.

NoteConsider the first sample. The first assumption actually coincides with the height remembered by Vasya. In the second assumption the heights of the trees are (4, 2, 3, 4), in the third one they are (1, 2, 3, 3) and in the fourth one they are (1, 2, 3, 5).


After a drawn-out mooclear arms race, Farmer John and the Mischievous Mess Makers have finally agreed to establish peace. They plan to divide the territory of Bovinia with a line passing through at least two of the n outposts scattered throughout the land. These outposts, remnants of the conflict, are located at the points (x1, y1), (x2, y2), ..., (xn, yn).In order to find the optimal dividing line, Farmer John and Elsie have plotted a map of Bovinia on the coordinate plane. Farmer John's farm and the Mischievous Mess Makers' base are located at the points P = (a, 0) and Q = ( - a, 0), respectively. Because they seek a lasting peace, Farmer John and Elsie would like to minimize the maximum difference between the distances from any point on the line to P and Q.Formally, define the difference of a line  relative to two points P and Q as the smallest real number d so that for all points X on line , |PX - QX| ≤ d. (It is guaranteed that d exists and is unique.) They wish to find the line  passing through two distinct outposts (xi, yi) and (xj, yj) such that the difference of  relative to P and Q is minimized.

The first line of the input contains two integers n and a (2 ≤ n ≤ 100 000, 1 ≤ a ≤ 10 000) — the number of outposts and the coordinates of the farm and the base, respectively. The following n lines describe the locations of the outposts as pairs of integers (xi, yi) (|xi|, |yi| ≤ 10 000). These points are distinct from each other as well as from P and Q.

Print a single real number—the difference of the optimal dividing line. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6.  Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .

NoteIn the first sample case, the only possible line  is y = x - 1. It can be shown that the point X which maximizes |PX - QX| is (13, 12), with , which is .In the second sample case, if we pick the points (0, 1) and (0,  - 3), we get  as x = 0. Because PX = QX on this line, the minimum possible difference is 0.


My name is James diGriz, I'm the most clever robber and treasure hunter in the whole galaxy. There are books written about my adventures and songs about my operations, though you were able to catch me up in a pretty awkward moment.I was able to hide from cameras, outsmart all the guards and pass numerous traps, but when I finally reached the treasure box and opened it, I have accidentally started the clockwork bomb! Luckily, I have met such kind of bombs before and I know that the clockwork mechanism can be stopped by connecting contacts with wires on the control panel of the bomb in a certain manner.I see n contacts connected by n - 1 wires. Contacts are numbered with integers from 1 to n. Bomb has a security mechanism that ensures the following condition: if there exist k ≥ 2 contacts c1, c2, ..., ck forming a circuit, i. e. there exist k distinct wires between contacts c1 and c2, c2 and c3, ..., ck and c1, then the bomb immediately explodes and my story ends here. In particular, if two contacts are connected by more than one wire they form a circuit of length 2. It is also prohibited to connect a contact with itself.On the other hand, if I disconnect more than one wire (i. e. at some moment there will be no more than n - 2 wires in the scheme) then the other security check fails and the bomb also explodes. So, the only thing I can do is to unplug some wire and plug it into a new place ensuring the fact that no circuits appear.I know how I should put the wires in order to stop the clockwork. But my time is running out! Help me get out of this alive: find the sequence of operations each of which consists of unplugging some wire and putting it into another place so that the bomb is defused.

The first line of the input contains n (2 ≤ n ≤ 500 000), the number of contacts. Each of the following n - 1 lines contains two of integers xi and yi (1 ≤ xi, yi ≤ n, xi ≠ yi) denoting the contacts currently connected by the i-th wire. The remaining n - 1 lines contain the description of the sought scheme in the same format. It is guaranteed that the starting and the ending schemes are correct (i. e. do not contain cicuits nor wires connecting contact with itself).

The first line should contain k (k ≥ 0) — the minimum number of moves of unplugging and plugging back some wire required to defuse the bomb. In each of the following k lines output four integers ai, bi, ci, di meaning that on the i-th step it is neccesary to unplug the wire connecting the contacts ai and bi and plug it to the contacts ci and di. Of course the wire connecting contacts ai and bi should be present in the scheme. If there is no correct sequence transforming the existing scheme into the sought one, output -1.

NotePicture with the clarification for the sample tests:


There are n pictures delivered for the new exhibition. The i-th painting has beauty ai. We know that a visitor becomes happy every time he passes from a painting to a more beautiful one.We are allowed to arranged pictures in any order. What is the maximum possible number of times the visitor may become happy while passing all pictures from first to last? In other words, we are allowed to rearrange elements of a in any order. What is the maximum possible number of indices i (1 ≤ i ≤ n - 1), such that ai + 1 > ai.

The first line of the input contains integer n (1 ≤ n ≤ 1000) — the number of painting. The second line contains the sequence a1, a2, ..., an (1 ≤ ai ≤ 1000), where ai means the beauty of the i-th painting.

Print one integer — the maximum possible number of neighbouring pairs, such that ai + 1 > ai, after the optimal rearrangement.

NoteIn the first sample, the optimal order is: 10, 20, 30, 40, 50.In the second sample, the optimal order is: 100, 200, 100, 200.


Friends are going to play console. They have two joysticks and only one charger for them. Initially first joystick is charged at a1 percent and second one is charged at a2 percent. You can connect charger to a joystick only at the beginning of each minute. In one minute joystick either discharges by 2 percent (if not connected to a charger) or charges by 1 percent (if connected to a charger).Game continues while both joysticks have a positive charge. Hence, if at the beginning of minute some joystick is charged by 1 percent, it has to be connected to a charger, otherwise the game stops. If some joystick completely discharges (its charge turns to 0), the game also stops.Determine the maximum number of minutes that game can last. It is prohibited to pause the game, i. e. at each moment both joysticks should be enabled. It is allowed for joystick to be charged by more than 100 percent.

The first line of the input contains two positive integers a1 and a2 (1 ≤ a1, a2 ≤ 100), the initial charge level of first and second joystick respectively.

Output the only integer, the maximum number of minutes that the game can last. Game continues until some joystick is discharged.

NoteIn the first sample game lasts for 6 minute by using the following algorithm:  at the beginning of the first minute connect first joystick to the charger, by the end of this minute first joystick is at 4%, second is at 3%;  continue the game without changing charger, by the end of the second minute the first joystick is at 5%, second is at 1%;  at the beginning of the third minute connect second joystick to the charger, after this minute the first joystick is at 3%, the second one is at 2%;  continue the game without changing charger, by the end of the fourth minute first joystick is at 1%, second one is at 3%;  at the beginning of the fifth minute connect first joystick to the charger, after this minute the first joystick is at 2%, the second one is at 1%;  at the beginning of the sixth minute connect second joystick to the charger, after this minute the first joystick is at 0%, the second one is at 2%. After that the first joystick is completely discharged and the game is stopped.


A student of z-school found a kind of sorting called z-sort. The array a with n elements are z-sorted if two conditions hold:  ai ≥ ai - 1 for all even i,  ai ≤ ai - 1 for all odd i > 1. For example the arrays [1,2,1,2] and [1,1,1,1] are z-sorted while the array [1,2,3,4] isn’t z-sorted.Can you make the array z-sorted?

The first line contains a single integer n (1 ≤ n ≤ 1000) — the number of elements in the array a. The second line contains n integers ai (1 ≤ ai ≤ 109) — the elements of the array a.

If it's possible to make the array a z-sorted print n space separated integers ai — the elements after z-sort. Otherwise print the only word "Impossible".


The 9-th grade student Gabriel noticed a caterpillar on a tree when walking around in a forest after the classes. The caterpillar was on the height h1 cm from the ground. On the height h2 cm (h2 > h1) on the same tree hung an apple and the caterpillar was crawling to the apple.Gabriel is interested when the caterpillar gets the apple. He noted that the caterpillar goes up by a cm per hour by day and slips down by b cm per hour by night.In how many days Gabriel should return to the forest to see the caterpillar get the apple. You can consider that the day starts at 10 am and finishes at 10 pm. Gabriel's classes finish at 2 pm. You can consider that Gabriel noticed the caterpillar just after the classes at 2 pm.Note that the forest is magic so the caterpillar can slip down under the ground and then lift to the apple.

The first line contains two integers h1, h2 (1 ≤ h1 < h2 ≤ 105) — the heights of the position of the caterpillar and the apple in centimeters. The second line contains two integers a, b (1 ≤ a, b ≤ 105) — the distance the caterpillar goes up by day and slips down by night, in centimeters per hour.

Print the only integer k — the number of days Gabriel should wait to return to the forest and see the caterpillar getting the apple. If the caterpillar can't get the apple print the only integer  - 1.

NoteIn the first example at 10 pm of the first day the caterpillar gets the height 26. At 10 am of the next day it slips down to the height 14. And finally at 6 pm of the same day the caterpillar gets the apple.Note that in the last example the caterpillar was slipping down under the ground and getting the apple on the next day.


You are given a permutation p of length n. Also you are given m foe pairs (ai, bi) (1 ≤ ai, bi ≤ n, ai ≠ bi). Your task is to count the number of different intervals (x, y) (1 ≤ x ≤ y ≤ n) that do not contain any foe pairs. So you shouldn't count intervals (x, y) that contain at least one foe pair in it (the positions and order of the values from the foe pair are not important).Consider some example: p = [1, 3, 2, 4] and foe pairs are {(3, 2), (4, 2)}. The interval (1, 3) is incorrect because it contains a foe pair (3, 2). The interval (1, 4) is also incorrect because it contains two foe pairs (3, 2) and (4, 2). But the interval (1, 2) is correct because it doesn't contain any foe pair.

The first line contains two integers n and m (1 ≤ n, m ≤ 3·105) — the length of the permutation p and the number of foe pairs. The second line contains n distinct integers pi (1 ≤ pi ≤ n) — the elements of the permutation p. Each of the next m lines contains two integers (ai, bi) (1 ≤ ai, bi ≤ n, ai ≠ bi) — the i-th foe pair. Note a foe pair can appear multiple times in the given list.

Print the only integer c — the number of different intervals (x, y) that does not contain any foe pairs. Note that the answer can be too large, so you should use 64-bit integer type to store it. In C++ you can use the long long integer type and in Java you can use long integer type.

NoteIn the first example the intervals from the answer are (1, 1), (1, 2), (2, 2), (3, 3) and (4, 4).


In an attempt to make peace with the Mischievious Mess Makers, Bessie and Farmer John are planning to plant some flower gardens to complement the lush, grassy fields of Bovinia. As any good horticulturist knows, each garden they plant must have the exact same arrangement of flowers. Initially, Farmer John has n different species of flowers he can plant, with ai flowers of the i-th species.On each of the next q days, Farmer John will receive a batch of flowers of a new species. On day j, he will receive cj flowers of the same species, but of a different species from those Farmer John already has. Farmer John, knowing the right balance between extravagance and minimalism, wants exactly k species of flowers to be used. Furthermore, to reduce waste, each flower of the k species Farmer John chooses must be planted in some garden. And each of the gardens must be identical; that is to say that each of the k chosen species should have an equal number of flowers in each garden. As Farmer John is a proponent of national equality, he would like to create the greatest number of gardens possible.After receiving flowers on each of these q days, Farmer John would like to know the sum, over all possible choices of k species, of the maximum number of gardens he could create. Since this could be a large number, you should output your result modulo 109 + 7.

The first line of the input contains three integers n, k and q (1 ≤ k ≤ n ≤ 100 000, 1 ≤ q ≤ 100 000). The i-th (1 ≤ i ≤ n) of the next n lines of the input contains an integer ai (1 ≤ ai ≤ 1 000 000), the number of flowers of species i Farmer John has initially. The j-th (1 ≤ j ≤ q) of the next q lines of the input contains an integer cj (1 ≤ cj ≤ 1 000 000), the number of flowers of a new species Farmer John receives on day j.

After each of the q days, output the sum of the maximum possible number of gardens, where the sum is taken over all possible choices of k species, modulo 109 + 7.

NoteIn the first sample case, after the first day Farmer John has (4, 6, 9, 8) of each type of flower, and k = 3.Choosing (4, 6, 8) lets him make 2 gardens, each with (2, 3, 4) of each flower, respectively. Choosing (4, 6, 9), (4, 9, 8) and (6, 9, 8) each only let him make one garden, since there is no number of gardens that each species can be evenly split into. So the sum over all choices of k = 3 flowers is 2 + 1 + 1 + 1 = 5.After the second day, Farmer John has (4, 6, 9, 8, 6) of each flower. The sum over all choices is 1 + 2 + 2 + 1 + 1 + 2 + 2 + 3 + 1 + 1 = 16.In the second sample case, k = 1. With x flowers Farmer John can make x gardens. So the answers to the queries are 6 + 5 + 4 + 3 + 2 = 20 and 6 + 5 + 4 + 3 + 2 + 1 = 21.


You are given n segments on a line. There are no ends of some segments that coincide. For each segment find the number of segments it contains.

The first line contains a single integer n (1 ≤ n ≤ 2·105) — the number of segments on a line. Each of the next n lines contains two integers li and ri ( - 109 ≤ li < ri ≤ 109) — the coordinates of the left and the right ends of the i-th segment. It is guaranteed that there are no ends of some segments that coincide.

Print n lines. The j-th of them should contain the only integer aj — the number of segments contained in the j-th segment.


Johnny is playing a well-known computer game. The game are in some country, where the player can freely travel, pass quests and gain an experience.In that country there are n islands and m bridges between them, so you can travel from any island to any other. In the middle of some bridges are lying ancient powerful artifacts. Johnny is not interested in artifacts, but he can get some money by selling some artifact.At the start Johnny is in the island a and the artifact-dealer is in the island b (possibly they are on the same island). Johnny wants to find some artifact, come to the dealer and sell it. The only difficulty is that bridges are too old and destroying right after passing over them. Johnnie's character can't swim, fly and teleport, so the problem became too difficult.Note that Johnny can't pass the half of the bridge, collect the artifact and return to the same island. Determine if Johnny can find some artifact and sell it.

The first line contains two integers n and m (1 ≤ n ≤ 3·105, 0 ≤ m ≤ 3·105) — the number of islands and bridges in the game. Each of the next m lines contains the description of the bridge — three integers xi, yi, zi (1 ≤ xi, yi ≤ n, xi ≠ yi, 0 ≤ zi ≤ 1), where xi and yi are the islands connected by the i-th bridge, zi equals to one if that bridge contains an artifact and to zero otherwise. There are no more than one bridge between any pair of islands. It is guaranteed that it's possible to travel between any pair of islands. The last line contains two integers a and b (1 ≤ a, b ≤ n) — the islands where are Johnny and the artifact-dealer respectively.

If Johnny can find some artifact and sell it print the only word "YES" (without quotes). Otherwise print the word "NO" (without quotes).


n ants are on a circle of length m. An ant travels one unit of distance per one unit of time. Initially, the ant number i is located at the position si and is facing in the direction di (which is either L or R). Positions are numbered in counterclockwise order starting from some point. Positions of the all ants are distinct.All the ants move simultaneously, and whenever two ants touch, they will both switch their directions. Note that it is possible for an ant to move in some direction for a half of a unit of time and in opposite direction for another half of a unit of time.Print the positions of the ants after t time units.

The first line contains three integers n, m and t (2 ≤ n ≤ 3·105, 2 ≤ m ≤ 109, 0 ≤ t ≤ 1018) — the number of ants, the length of the circle and the number of time units. Each of the next n lines contains integer si and symbol di (1 ≤ si ≤ m and di is either L or R) — the position and the direction of the i-th ant at the start. The directions L and R corresponds to the clockwise and counterclockwise directions, respectively. It is guaranteed that all positions si are distinct.

Print n integers xj — the position of the j-th ant after t units of time. The ants are numbered from 1 to n in order of their appearing in input.


Limak is a little polar bear. He has n balls, the i-th ball has size ti.Limak wants to give one ball to each of his three friends. Giving gifts isn't easy — there are two rules Limak must obey to make friends happy:  No two friends can get balls of the same size.  No two friends can get balls of sizes that differ by more than 2. For example, Limak can choose balls with sizes 4, 5 and 3, or balls with sizes 90, 91 and 92. But he can't choose balls with sizes 5, 5 and 6 (two friends would get balls of the same size), and he can't choose balls with sizes 30, 31 and 33 (because sizes 30 and 33 differ by more than 2).Your task is to check whether Limak can choose three balls that satisfy conditions above.

The first line of the input contains one integer n (3 ≤ n ≤ 50) — the number of balls Limak has. The second line contains n integers t1, t2, ..., tn (1 ≤ ti ≤ 1000) where ti denotes the size of the i-th ball.

Print "YES" (without quotes) if Limak can choose three balls of distinct sizes, such that any two of them differ by no more than 2. Otherwise, print "NO" (without quotes).

NoteIn the first sample, there are 4 balls and Limak is able to choose three of them to satisfy the rules. He must must choose balls with sizes 18, 16 and 17.In the second sample, there is no way to give gifts to three friends without breaking the rules.In the third sample, there is even more than one way to choose balls:  Choose balls with sizes 3, 4 and 5.  Choose balls with sizes 972, 970, 971.


Limak is a little polar bear. Polar bears hate long strings and thus they like to compress them. You should also know that Limak is so young that he knows only first six letters of the English alphabet: 'a', 'b', 'c', 'd', 'e' and 'f'.You are given a set of q possible operations. Limak can perform them in any order, any operation may be applied any number of times. The i-th operation is described by a string ai of length two and a string bi of length one. No two of q possible operations have the same string ai.When Limak has a string s he can perform the i-th operation on s if the first two letters of s match a two-letter string ai. Performing the i-th operation removes first two letters of s and inserts there a string bi. See the notes section for further clarification.You may note that performing an operation decreases the length of a string s exactly by 1. Also, for some sets of operations there may be a string that cannot be compressed any further, because the first two letters don't match any ai.Limak wants to start with a string of length n and perform n - 1 operations to finally get a one-letter string "a". In how many ways can he choose the starting string to be able to get "a"? Remember that Limak can use only letters he knows.

The first line contains two integers n and q (2 ≤ n ≤ 6, 1 ≤ q ≤ 36) — the length of the initial string and the number of available operations. The next q lines describe the possible operations. The i-th of them contains two strings ai and bi (|ai| = 2, |bi| = 1). It's guaranteed that ai ≠ aj for i ≠ j and that all ai and bi consist of only first six lowercase English letters.

Print the number of strings of length n that Limak will be able to transform to string "a" by applying only operations given in the input.

NoteIn the first sample, we count initial strings of length 3 from which Limak can get a required string "a". There are 4 such strings: "abb", "cab", "cca", "eea". The first one Limak can compress using operation 1 two times (changing "ab" to a single "a"). The first operation would change "abb" to "ab" and the second operation would change "ab" to "a".Other three strings may be compressed as follows:   "cab"  "ab"  "a"  "cca"  "ca"  "a"  "eea"  "ca"  "a" In the second sample, the only correct initial string is "eb" because it can be immediately compressed to "a".

2 8
af e
dc d
cc f
bc b
da b
eb a
bb b
ff c


Niwel is a little golden bear. As everyone knows, bears live in forests, but Niwel got tired of seeing all the trees so he decided to move to the city.In the city, Niwel took on a job managing bears to deliver goods. The city that he lives in can be represented as a directed graph with n nodes and m edges. Each edge has a weight capacity. A delivery consists of a bear carrying weights with their bear hands on a simple path from node 1 to node n. The total weight that travels across a particular edge must not exceed the weight capacity of that edge.Niwel has exactly x bears. In the interest of fairness, no bear can rest, and the weight that each bear carries must be exactly the same. However, each bear may take different paths if they like.Niwel would like to determine, what is the maximum amount of weight he can deliver (it's the sum of weights carried by bears). Find the maximum weight.

The first line contains three integers n, m and x (2 ≤ n ≤ 50, 1 ≤ m ≤ 500, 1 ≤ x ≤ 100 000) — the number of nodes, the number of directed edges and the number of bears, respectively. Each of the following m lines contains three integers ai, bi and ci (1 ≤ ai, bi ≤ n, ai ≠ bi, 1 ≤ ci ≤ 1 000 000). This represents a directed edge from node ai to bi with weight capacity ci. There are no self loops and no multiple edges from one city to the other city. More formally, for each i and j that i ≠ j it's guaranteed that ai ≠ aj or bi ≠ bj. It is also guaranteed that there is at least one path from node 1 to node n.

Print one real value on a single line — the maximum amount of weight Niwel can deliver if he uses exactly x bears. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6. Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct if .

NoteIn the first sample, Niwel has three bears. Two bears can choose the path , while one bear can choose the path . Even though the bear that goes on the path  can carry one unit of weight, in the interest of fairness, he is restricted to carry 0.5 units of weight. Thus, the total weight is 1.5 units overall. Note that even though Niwel can deliver more weight with just 2 bears, he must use exactly 3 bears on this day.