Vasya has the square chessboard of size n × n and m rooks. Initially the chessboard is empty. Vasya will consequently put the rooks on the board one after another.The cell of the field is under rook's attack, if there is at least one rook located in the same row or in the same column with this cell. If there is a rook located in the cell, this cell is also under attack.You are given the positions of the board where Vasya will put rooks. For each rook you have to determine the number of cells which are not under attack after Vasya puts it on the board.

The first line of the input contains two integers n and m (1 ≤ n ≤ 100 000, 1 ≤ m ≤ min(100 000, n2)) — the size of the board and the number of rooks.  Each of the next m lines contains integers xi and yi (1 ≤ xi, yi ≤ n) — the number of the row and the number of the column where Vasya will put the i-th rook. Vasya puts rooks on the board in the order they appear in the input. It is guaranteed that any cell will contain no more than one rook.

Print m integer, the i-th of them should be equal to the number of cells that are not under attack after first i rooks are put.

NoteOn the picture below show the state of the board after put each of the three rooks. The cells which painted with grey color is not under the attack.


There are n cards (n is even) in the deck. Each card has a positive integer written on it. n / 2 people will play new card game. At the beginning of the game each player gets two cards, each card is given to exactly one player. Find the way to distribute cards such that the sum of values written of the cards will be equal for each player. It is guaranteed that it is always possible.

The first line of the input contains integer n (2 ≤ n ≤ 100) — the number of cards in the deck. It is guaranteed that n is even. The second line contains the sequence of n positive integers a1, a2, ..., an (1 ≤ ai ≤ 100), where ai is equal to the number written on the i-th card.

Print n / 2 pairs of integers, the i-th pair denote the cards that should be given to the i-th player. Each card should be given to exactly one player. Cards are numbered in the order they appear in the input. It is guaranteed that solution exists. If there are several correct answers, you are allowed to print any of them.

NoteIn the first sample, cards are distributed in such a way that each player has the sum of numbers written on his cards equal to 8. In the second sample, all values ai are equal. Thus, any distribution is acceptable.


Theseus has just arrived to Crete to fight Minotaur. He found a labyrinth that has a form of a rectangular field of size n × m and consists of blocks of size 1 × 1.Each block of the labyrinth has a button that rotates all blocks 90 degrees clockwise. Each block rotates around its center and doesn't change its position in the labyrinth. Also, each block has some number of doors (possibly none). In one minute, Theseus can either push the button in order to rotate all the blocks 90 degrees clockwise or pass to the neighbouring block. Theseus can go from block A to some neighbouring block B only if block A has a door that leads to block B and block B has a door that leads to block A.Theseus found an entrance to labyrinth and is now located in block (xT, yT) — the block in the row xT and column yT. Theseus know that the Minotaur is hiding in block (xM, yM) and wants to know the minimum number of minutes required to get there.Theseus is a hero, not a programmer, so he asks you to help him.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 1000) — the number of rows and the number of columns in labyrinth, respectively. Each of the following n lines contains m characters, describing the blocks of the labyrinth. The possible characters are:   «+» means this block has 4 doors (one door to each neighbouring block);  «-» means this block has 2 doors — to the left and to the right neighbours;  «|» means this block has 2 doors — to the top and to the bottom neighbours;  «^» means this block has 1 door — to the top neighbour;  «>» means this block has 1 door — to the right neighbour;  «<» means this block has 1 door — to the left neighbour;  «v» means this block has 1 door — to the bottom neighbour; «L» means this block has 3 doors — to all neighbours except left one;  «R» means this block has 3 doors — to all neighbours except right one;  «U» means this block has 3 doors — to all neighbours except top one;  «D» means this block has 3 doors — to all neighbours except bottom one; «*» means this block is a wall and has no doors.  Left, right, top and bottom are defined from representing labyrinth as a table, where rows are numbered from 1 to n from top to bottom and columns are numbered from 1 to m from left to right. Next line contains two integers — coordinates of the block (xT, yT) (1 ≤ xT ≤ n, 1 ≤ yT ≤ m), where Theseus is initially located. Last line contains two integers — coordinates of the block (xM, yM) (1 ≤ xM ≤ n, 1 ≤ yM ≤ m), where Minotaur hides. It's guaranteed that both the block where Theseus starts and the block where Minotaur is hiding have at least one door. Theseus and Minotaur may be initially located at the same block.

If Theseus is not able to get to Minotaur, then print -1 in the only line of the output. Otherwise, print the minimum number of minutes required to get to the block where Minotaur is hiding.

NoteAssume that Theseus starts at the block (xT, yT) at the moment 0.


Bomboslav set up a branding agency and now helps companies to create new logos and advertising slogans. In term of this problems, slogan of the company should be a non-empty substring of its name. For example, if the company name is "hornsandhoofs", then substrings "sand" and "hor" could be its slogans, while strings "e" and "hornss" can not.Sometimes the company performs rebranding and changes its slogan. Slogan A is considered to be cooler than slogan B if B appears in A as a substring at least twice (this occurrences are allowed to overlap). For example, slogan A =  "abacaba" is cooler than slogan B =  "ba", slogan A =  "abcbcbe" is cooler than slogan B =  "bcb", but slogan A =  "aaaaaa" is not cooler than slogan B =  "aba".You are given the company name w and your task is to help Bomboslav determine the length of the longest sequence of slogans s1, s2, ..., sk, such that any slogan in the sequence is cooler than the previous one.

The first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the length of the company name that asks Bomboslav to help. The second line contains the string w of length n, that consists of lowercase English letters.

Print a single integer — the maximum possible length of the sequence of slogans of the company named w, such that any slogan in the sequence (except the first one) is cooler than the previous


You are given n integers a1, a2, ..., an. Find the number of pairs of indexes i, j (i < j) that ai + aj is a power of 2 (i. e. some integer x exists so that ai + aj = 2x).

The first line contains the single positive integer n (1 ≤ n ≤ 105) — the number of integers. The second line contains n positive integers a1, a2, ..., an (1 ≤ ai ≤ 109).

Print the number of pairs of indexes i, j (i < j) that ai + aj is a power of 2.

NoteIn the first example the following pairs of indexes include in answer: (1, 4) and (2, 4).In the second example all pairs of indexes (i, j) (where i < j) include in answer.


You are given array consisting of n integers. Your task is to find the maximum length of an increasing subarray of the given array.A subarray is the sequence of consecutive elements of the array. Subarray is called increasing if each element of this subarray strictly greater than previous.

The first line contains single positive integer n (1 ≤ n ≤ 105) — the number of integers. The second line contains n positive integers a1, a2, ..., an (1 ≤ ai ≤ 109).

Print the maximum length of an increasing subarray of the given array.


You are given a functional graph. It is a directed graph, in which from each vertex goes exactly one arc. The vertices are numerated from 0 to n - 1.Graph is given as the array f0, f1, ..., fn - 1, where fi — the number of vertex to which goes the only arc from the vertex i. Besides you are given array with weights of the arcs w0, w1, ..., wn - 1, where wi — the arc weight from i to fi.    The graph from the first sample test. Also you are given the integer k (the length of the path) and you need to find for each vertex two numbers si and mi, where:  si — the sum of the weights of all arcs of the path with length equals to k which starts from the vertex i;  mi — the minimal weight from all arcs on the path with length k which starts from the vertex i. The length of the path is the number of arcs on this path.

The first line contains two integers n, k (1 ≤ n ≤ 105, 1 ≤ k ≤ 1010). The second line contains the sequence f0, f1, ..., fn - 1 (0 ≤ fi < n) and the third — the sequence w0, w1, ..., wn - 1 (0 ≤ wi ≤ 108).

Print n lines, the pair of integers si, mi in each line.


You are given n points on the straight line — the positions (x-coordinates) of the cities and m points on the same line — the positions (x-coordinates) of the cellular towers. All towers work in the same way — they provide cellular network for all cities, which are located at the distance which is no more than r from this tower.Your task is to find minimal r that each city has been provided by cellular network, i.e. for each city there is at least one cellular tower at the distance which is no more than r.If r = 0 then a tower provides cellular network only for the point where it is located. One tower can provide cellular network for any number of cities, but all these cities must be at the distance which is no more than r from this tower.

The first line contains two positive integers n and m (1 ≤ n, m ≤ 105) — the number of cities and the number of cellular towers. The second line contains a sequence of n integers a1, a2, ..., an ( - 109 ≤ ai ≤ 109) — the coordinates of cities. It is allowed that there are any number of cities in the same point. All coordinates ai are given in non-decreasing order. The third line contains a sequence of m integers b1, b2, ..., bm ( - 109 ≤ bj ≤ 109) — the coordinates of cellular towers. It is allowed that there are any number of towers in the same point. All coordinates bj are given in non-decreasing order.

Print minimal r so that each city will be covered by cellular network.


Alice wants to send an important message to Bob. Message a = (a1, ..., an) is a sequence of positive integers (characters).To compress the message Alice wants to use binary Huffman coding. We recall that binary Huffman code, or binary prefix code is a function f, that maps each letter that appears in the string to some binary string (that is, string consisting of characters '0' and '1' only) such that for each pair of different characters ai and aj string f(ai) is not a prefix of f(aj) (and vice versa). The result of the encoding of the message a1, a2, ..., an is the concatenation of the encoding of each character, that is the string f(a1)f(a2)... f(an). Huffman codes are very useful, as the compressed message can be easily and uniquely decompressed, if the function f is given. Code is usually chosen in order to minimize the total length of the compressed message, i.e. the length of the string f(a1)f(a2)... f(an).Because of security issues Alice doesn't want to send the whole message. Instead, she picks some substrings of the message and wants to send them separately. For each of the given substrings ali... ari she wants to know the minimum possible length of the Huffman coding. Help her solve this problem.

The first line of the input contains the single integer n (1 ≤ n ≤ 100 000) — the length of the initial message. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 100 000) — characters of the message. Next line contains the single integer q (1 ≤ q ≤ 100 000) — the number of queries. Then follow q lines with queries descriptions. The i-th of these lines contains two integers li and ri (1 ≤ li ≤ ri ≤ n) — the position of the left and right ends of the i-th substring respectively. Positions are numbered from 1. Substrings may overlap in any way. The same substring may appear in the input more than once.

Print q lines. Each line should contain a single integer — the minimum possible length of the Huffman encoding of the substring ali... ari.

NoteIn the first query, one of the optimal ways to encode the substring is to map 1 to "0", 2 to "10" and 3 to "11".Note that it is correct to map the letter to the empty substring (as in the fifth query from the sample).


Vasiliy has a car and he wants to get from home to the post office. The distance which he needs to pass equals to d kilometers.Vasiliy's car is not new — it breaks after driven every k kilometers and Vasiliy needs t seconds to repair it. After repairing his car Vasiliy can drive again (but after k kilometers it will break again, and so on). In the beginning of the trip the car is just from repair station.To drive one kilometer on car Vasiliy spends a seconds, to walk one kilometer on foot he needs b seconds (a < b).Your task is to find minimal time after which Vasiliy will be able to reach the post office. Consider that in every moment of time Vasiliy can left his car and start to go on foot.

The first line contains 5 positive integers d, k, a, b, t (1 ≤ d ≤ 1012; 1 ≤ k, a, b, t ≤ 106; a < b), where:   d — the distance from home to the post office;  k — the distance, which car is able to drive before breaking;  a — the time, which Vasiliy spends to drive 1 kilometer on his car;  b — the time, which Vasiliy spends to walk 1 kilometer on foot;  t — the time, which Vasiliy spends to repair his car.

Print the minimal time after which Vasiliy will be able to reach the post office.

NoteIn the first example Vasiliy needs to drive the first 2 kilometers on the car (in 2 seconds) and then to walk on foot 3 kilometers (in 12 seconds). So the answer equals to 14 seconds.In the second example Vasiliy needs to drive the first 2 kilometers on the car (in 2 seconds), then repair his car (in 5 seconds) and drive 2 kilometers more on the car (in 2 seconds). After that he needs to walk on foot 1 kilometer (in 4 seconds). So the answer equals to 13 seconds.


Sergei B., the young coach of Pokemons, has found the big house which consists of n flats ordered in a row from left to right. It is possible to enter each flat from the street. It is possible to go out from each flat. Also, each flat is connected with the flat to the left and the flat to the right. Flat number 1 is only connected with the flat number 2 and the flat number n is only connected with the flat number n - 1.There is exactly one Pokemon of some type in each of these flats. Sergei B. asked residents of the house to let him enter their flats in order to catch Pokemons. After consulting the residents of the house decided to let Sergei B. enter one flat from the street, visit several flats and then go out from some flat. But they won't let him visit the same flat more than once. Sergei B. was very pleased, and now he wants to visit as few flats as possible in order to collect Pokemons of all types that appear in this house. Your task is to help him and determine this minimum number of flats he has to visit.

The first line contains the integer n (1 ≤ n ≤ 100 000) — the number of flats in the house. The second line contains the row s with the length n, it consists of uppercase and lowercase letters of English alphabet, the i-th letter equals the type of Pokemon, which is in the flat number i.

Print the minimum number of flats which Sergei B. should visit in order to catch Pokemons of all types which there are in the house.

NoteIn the first test Sergei B. can begin, for example, from the flat number 1 and end in the flat number 2.In the second test Sergei B. can begin, for example, from the flat number 4 and end in the flat number 6. In the third test Sergei B. must begin from the flat number 2 and end in the flat number 6.


Mishka is a little polar bear. As known, little bears loves spending their free time playing dice for chocolates. Once in a wonderful sunny morning, walking around blocks of ice, Mishka met her friend Chris, and they started playing the game.Rules of the game are very simple: at first number of rounds n is defined. In every round each of the players throws a cubical dice with distinct numbers from 1 to 6 written on its faces. Player, whose value after throwing the dice is greater, wins the round. In case if player dice values are equal, no one of them is a winner.In average, player, who won most of the rounds, is the winner of the game. In case if two players won the same number of rounds, the result of the game is draw.Mishka is still very little and can't count wins and losses, so she asked you to watch their game and determine its result. Please help her!

The first line of the input contains single integer n n (1 ≤ n ≤ 100) — the number of game rounds. The next n lines contains rounds description. i-th of them contains pair of integers mi and ci (1 ≤ mi,  ci ≤ 6) — values on dice upper face after Mishka's and Chris' throws in i-th round respectively.

If Mishka is the winner of the game, print "Mishka" (without quotes) in the only line. If Chris is the winner of the game, print "Chris" (without quotes) in the only line. If the result of the game is draw, print "Friendship is magic!^^" (without quotes) in the only line.

NoteIn the first sample case Mishka loses the first round, but wins second and third rounds and thus she is the winner of the game.In the second sample case Mishka wins the first round, Chris wins the second round, and the game ends with draw with score 1:1.In the third sample case Chris wins the first round, but there is no winner of the next two rounds. The winner of the game is Chris.


The big consignment of t-shirts goes on sale in the shop before the beginning of the spring. In all n types of t-shirts go on sale. The t-shirt of the i-th type has two integer parameters — ci and qi, where ci — is the price of the i-th type t-shirt, qi — is the quality of the i-th type t-shirt. It should be assumed that the unlimited number of t-shirts of each type goes on sale in the shop, but in general the quality is not concerned with the price. As predicted, k customers will come to the shop within the next month, the j-th customer will get ready to spend up to bj on buying t-shirts. All customers have the same strategy. First of all, the customer wants to buy the maximum possible number of the highest quality t-shirts, then to buy the maximum possible number of the highest quality t-shirts from residuary t-shirts and so on. At the same time among several same quality t-shirts the customer will buy one that is cheaper. The customers don't like the same t-shirts, so each customer will not buy more than one t-shirt of one type. Determine the number of t-shirts which each customer will buy, if they use the described strategy. All customers act independently from each other, and the purchase of one does not affect the purchase of another.

The first line contains the positive integer n (1 ≤ n ≤ 2·105) — the number of t-shirt types.  Each of the following n lines contains two integers ci and qi (1 ≤ ci, qi ≤ 109) — the price and the quality of the i-th type t-shirt. The next line contains the positive integer k (1 ≤ k ≤ 2·105) — the number of the customers.  The next line contains k positive integers b1, b2, ..., bk (1 ≤ bj ≤ 109), where the j-th number is equal to the sum, which the j-th customer gets ready to spend on t-shirts.

The first line of the input data should contain the sequence of k integers, where the i-th number should be equal to the number of t-shirts, which the i-th customer will buy.

NoteIn the first example the first customer will buy the t-shirt of the second type, then the t-shirt of the first type. He will spend 10 and will not be able to buy the t-shirt of the third type because it costs 4, and the customer will owe only 3. The second customer will buy all three t-shirts (at first, the t-shirt of the second type, then the t-shirt of the first type, and then the t-shirt of the third type). He will spend all money on it.


Little Mishka is a great traveller and she visited many countries. After thinking about where to travel this time, she chose XXX — beautiful, but little-known northern country.Here are some interesting facts about XXX:  XXX consists of n cities, k of whose (just imagine!) are capital cities.  All of cities in the country are beautiful, but each is beautiful in its own way. Beauty value of i-th city equals to ci.  All the cities are consecutively connected by the roads, including 1-st and n-th city, forming a cyclic route 1 — 2 — ... — n — 1. Formally, for every 1 ≤ i < n there is a road between i-th and i + 1-th city, and another one between 1-st and n-th city.  Each capital city is connected with each other city directly by the roads. Formally, if city x is a capital city, then for every 1 ≤ i ≤ n,  i ≠ x, there is a road between cities x and i.  There is at most one road between any two cities.  Price of passing a road directly depends on beauty values of cities it connects. Thus if there is a road between cities i and j, price of passing it equals ci·cj.Mishka started to gather her things for a trip, but didn't still decide which route to follow and thus she asked you to help her determine summary price of passing each of the roads in XXX. Formally, for every pair of cities a and b (a < b), such that there is a road between a and b you are to find sum of products ca·cb. Will you help her?

The first line of the input contains two integers n and k (3 ≤ n ≤ 100 000, 1 ≤ k ≤ n) — the number of cities in XXX and the number of capital cities among them. The second line of the input contains n integers c1, c2, ..., cn (1 ≤ ci ≤ 10 000) — beauty values of the cities. The third line of the input contains k distinct integers id1, id2, ..., idk (1 ≤ idi ≤ n) — indices of capital cities. Indices are given in ascending order.

Print the only integer — summary price of passing each of the roads in XXX.

NoteThis image describes first sample case:It is easy to see that summary price is equal to 17.This image describes second sample case:It is easy to see that summary price is equal to 71.


Thor is getting used to the Earth. As a gift Loki gave him a smartphone. There are n applications on this phone. Thor is fascinated by this phone. He has only one minor issue: he can't count the number of unread notifications generated by those applications (maybe Loki put a curse on it so he can't).q events are about to happen (in chronological order). They are of three types:  Application x generates a notification (this new notification is unread).  Thor reads all notifications generated so far by application x (he may re-read some notifications).  Thor reads the first t notifications generated by phone applications (notifications generated in first t events of the first type). It's guaranteed that there were at least t events of the first type before this event. Please note that he doesn't read first t unread notifications, he just reads the very first t notifications generated on his phone and he may re-read some of them in this operation. Please help Thor and tell him the number of unread notifications after each event. You may assume that initially there are no notifications in the phone.

The first line of input contains two integers n and q (1 ≤ n, q ≤ 300 000) — the number of applications and the number of events to happen. The next q lines contain the events. The i-th of these lines starts with an integer typei — type of the i-th event. If typei = 1 or typei = 2 then it is followed by an integer xi. Otherwise it is followed by an integer ti (1 ≤ typei ≤ 3, 1 ≤ xi ≤ n, 1 ≤ ti ≤ q).

Print the number of unread notifications after each event.

NoteIn the first sample:  Application 3 generates a notification (there is 1 unread notification).  Application 1 generates a notification (there are 2 unread notifications).  Application 2 generates a notification (there are 3 unread notifications).  Thor reads the notification generated by application 3, there are 2 unread notifications left. In the second sample test:  Application 2 generates a notification (there is 1 unread notification).  Application 4 generates a notification (there are 2 unread notifications).  Application 2 generates a notification (there are 3 unread notifications).  Thor reads first three notifications and since there are only three of them so far, there will be no unread notification left.  Application 3 generates a notification (there is 1 unread notification).  Application 3 generates a notification (there are 2 unread notifications).


Little Mishka enjoys programming. Since her birthday has just passed, her friends decided to present her with array of non-negative integers a1, a2, ..., an of n elements!Mishka loved the array and she instantly decided to determine its beauty value, but she is too little and can't process large arrays. Right because of that she invited you to visit her and asked you to process m queries.Each query is processed in the following way:  Two integers l and r (1 ≤ l ≤ r ≤ n) are specified — bounds of query segment.  Integers, presented in array segment [l,  r] (in sequence of integers al, al + 1, ..., ar) even number of times, are written down.  XOR-sum of written down integers is calculated, and this value is the answer for a query. Formally, if integers written down in point 2 are x1, x2, ..., xk, then Mishka wants to know the value , where  — operator of exclusive bitwise OR. Since only the little bears know the definition of array beauty, all you are to do is to answer each of queries presented.

The first line of the input contains single integer n (1 ≤ n ≤ 1 000 000) — the number of elements in the array. The second line of the input contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — array elements. The third line of the input contains single integer m (1 ≤ m ≤ 1 000 000) — the number of queries. Each of the next m lines describes corresponding query by a pair of integers l and r (1 ≤ l ≤ r ≤ n) — the bounds of query segment.

Print m non-negative integers — the answers for the queries in the order they appear in the input.

NoteIn the second sample:There is no integers in the segment of the first query, presented even number of times in the segment — the answer is 0.In the second query there is only integer 3 is presented even number of times — the answer is 3.In the third query only integer 1 is written down — the answer is 1.In the fourth query all array elements are considered. Only 1 and 2 are presented there even number of times. The answer is .In the fifth query 1 and 3 are written down. The answer is .


And while Mishka is enjoying her trip...Chris is a little brown bear. No one knows, where and when he met Mishka, but for a long time they are together (excluding her current trip). However, best friends are important too. John is Chris' best friend.Once walking with his friend, John gave Chris the following problem:At the infinite horizontal road of width w, bounded by lines y = 0 and y = w, there is a bus moving, presented as a convex polygon of n vertices. The bus moves continuously with a constant speed of v in a straight Ox line in direction of decreasing x coordinates, thus in time only x coordinates of its points are changing. Formally, after time t each of x coordinates of its points will be decreased by vt.There is a pedestrian in the point (0, 0), who can move only by a vertical pedestrian crossing, presented as a segment connecting points (0, 0) and (0, w) with any speed not exceeding u. Thus the pedestrian can move only in a straight line Oy in any direction with any speed not exceeding u and not leaving the road borders. The pedestrian can instantly change his speed, thus, for example, he can stop instantly.Please look at the sample note picture for better understanding.We consider the pedestrian is hit by the bus, if at any moment the point he is located in lies strictly inside the bus polygon (this means that if the point lies on the polygon vertex or on its edge, the pedestrian is not hit by the bus).You are given the bus position at the moment 0. Please help Chris determine minimum amount of time the pedestrian needs to cross the road and reach the point (0, w) and not to be hit by the bus.

The first line of the input contains four integers n, w, v, u (3 ≤ n ≤ 10 000, 1 ≤ w ≤ 109, 1 ≤ v,  u ≤ 1000) — the number of the bus polygon vertices, road width, bus speed and pedestrian speed respectively. The next n lines describes polygon vertices in counter-clockwise order. i-th of them contains pair of integers xi and yi ( - 109 ≤ xi ≤ 109, 0 ≤ yi ≤ w) — coordinates of i-th polygon point. It is guaranteed that the polygon is non-degenerate.

Print the single real t — the time the pedestrian needs to croos the road and not to be hit by the bus. The answer is considered correct if its relative or absolute error doesn't exceed 10 - 6.

NoteFollowing image describes initial position in the first sample case:


Scott Lang is at war with Darren Cross. There are n chairs in a hall where they are, numbered with 1, 2, ..., n from left to right. The i-th chair is located at coordinate xi. Scott is on chair number s and Cross is on chair number e. Scott can jump to all other chairs (not only neighboring chairs). He wants to start at his position (chair number s), visit each chair exactly once and end up on chair number e with Cross. As we all know, Scott can shrink or grow big (grow big only to his normal size), so at any moment of time he can be either small or large (normal). The thing is, he can only shrink or grow big while being on a chair (not in the air while jumping to another chair). Jumping takes time, but shrinking and growing big takes no time. Jumping from chair number i to chair number j takes |xi - xj| seconds. Also, jumping off a chair and landing on a chair takes extra amount of time. If Scott wants to jump to a chair on his left, he can only be small, and if he wants to jump to a chair on his right he should be large.Jumping off the i-th chair takes:  ci extra seconds if he's small.  di extra seconds otherwise (he's large). Also, landing on i-th chair takes:  bi extra seconds if he's small.  ai extra seconds otherwise (he's large). In simpler words, jumping from i-th chair to j-th chair takes exactly:  |xi - xj| + ci + bj seconds if j < i.  |xi - xj| + di + aj seconds otherwise (j > i). Given values of x, a, b, c, d find the minimum time Scott can get to Cross, assuming he wants to visit each chair exactly once.

The first line of the input contains three integers n, s and e (2 ≤ n ≤ 5000, 1 ≤ s, e ≤ n, s ≠ e) — the total number of chairs, starting and ending positions of Scott. The second line contains n integers x1, x2, ..., xn (1 ≤ x1 < x2 < ... < xn ≤ 109). The third line contains n integers a1, a2, ..., an (1 ≤ a1, a2, ..., an ≤ 109). The fourth line contains n integers b1, b2, ..., bn (1 ≤ b1, b2, ..., bn ≤ 109). The fifth line contains n integers c1, c2, ..., cn (1 ≤ c1, c2, ..., cn ≤ 109). The sixth line contains n integers d1, d2, ..., dn (1 ≤ d1, d2, ..., dn ≤ 109).

Print the minimum amount of time Scott needs to get to the Cross while visiting each chair exactly once.

NoteIn the sample testcase, an optimal solution would be . Spent time would be 17 + 24 + 23 + 20 + 33 + 22 = 139.


Steve Rogers is fascinated with new vibranium shields S.H.I.E.L.D gave him. They're all uncolored. There are n shields in total, the i-th shield is located at point (xi, yi) of the coordinate plane. It's possible that two or more shields share the same location.Steve wants to paint all these shields. He paints each shield in either red or blue. Painting a shield in red costs r dollars while painting it in blue costs b dollars.Additionally, there are m constraints Steve wants to be satisfied. The i-th constraint is provided by three integers ti, li and di:  If ti = 1, then the absolute difference between the number of red and blue shields on line x = li should not exceed di.  If ti = 2, then the absolute difference between the number of red and blue shields on line y = li should not exceed di. Steve gave you the task of finding the painting that satisfies all the condition and the total cost is minimum.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100 000) — the number of shields and the number of constraints respectively. The second line contains two integers r and b (1 ≤ r, b ≤ 109). The next n lines contain the shields coordinates. The i-th of these lines contains two integers xi and yi (1 ≤ xi, yi ≤ 109). The next m lines contain the constrains. The j-th of these lines contains three integers tj, lj and dj (1 ≤ tj ≤ 2, 1 ≤ lj ≤ 109, 0 ≤ dj ≤ n).

If satisfying all the constraints is impossible print -1 in first and only line of the output. Otherwise, print the minimum total cost in the first line of output. In the second line print a string of length n consisting of letters 'r' and 'b' only. The i-th character should be 'r' if the i-th shield should be painted red in the optimal answer and 'b' if it should be painted blue. The cost of painting shields in these colors should be equal the minimum cost you printed on the first line. If there exist more than one optimal solution, print any of them.


Natalia Romanova is trying to test something on the new gun S.H.I.E.L.D gave her. In order to determine the result of the test, she needs to find the number of answers to a certain equation. The equation is of form:Where  represents logical OR and  represents logical exclusive OR (XOR), and vi, j are some boolean variables or their negations. Natalia calls the left side of the equation a XNF formula. Each statement in brackets is called a clause, and vi, j are called literals.In the equation Natalia has, the left side is actually a 2-XNF-2 containing variables x1, x2, ..., xm and their negations. An XNF formula is 2-XNF-2 if:  For each 1 ≤ i ≤ n, ki ≤ 2, i.e. the size of each clause doesn't exceed two.  Each variable occurs in the formula at most two times (with negation and without negation in total). Please note that it's possible that a variable occurs twice but its negation doesn't occur in any clause (or vice versa). Natalia is given a formula of m variables, consisting of n clauses. Please, make sure to check the samples in order to properly understand how the formula looks like.Natalia is more into fight than theory, so she asked you to tell her the number of answers to this equation. More precisely, you need to find the number of ways to set x1, ..., xm with true and false (out of total of 2m ways) so that the equation is satisfied. Since this number can be extremely large, you need to print the answer modulo 109 + 7.Please, note that some variable may appear twice in one clause, or not appear in the equation at all (but still, setting it to false or true gives different ways to set variables).

The first line of input contains two integers n and m (1 ≤ n, m ≤ 100 000) — the number of clauses and the number of variables respectively. The next n lines contain the formula. The i-th of them starts with an integer ki — the number of literals in the i-th clause. It is followed by ki non-zero integers ai, 1, ..., ai, ki. If ai, j > 0 then vi, j is xai, j otherwise it's negation of x - ai, j (1 ≤ ki ≤ 2,  - m ≤ ai, j ≤ m, ai, j ≠ 0).

Print the answer modulo 1 000 000 007 (109 + 7) in one line.

NoteThe equation in the first sample is:The equation in the second sample is:The equation in the third sample is:


After playing with her beautiful array, Mishka decided to learn some math. After learning how to multiply, divide and what is divisibility, she is now interested in solving the following problem.You are given integer k and array a1, a2, ..., an of n integers. You are to find non-empty subsequence of array elements such that the product of its elements is divisible by k and it contains minimum possible number of elements.Formally, you are to find a sequence of indices 1 ≤ i1 < i2 < ... < im ≤ n such that  is divisible by k while m is minimum possible among all such variants.If there are more than one such subsequences, you should choose one among them, such that sum of its elements is minimum possible.Mishka quickly solved this problem. Will you do so?

The first line of the input contains two integers n and k (1 ≤ n ≤ 1 000, 1 ≤ k ≤ 1012). The second line of the input contains n integers a1, a2, ..., an (1 ≤ ai ≤ 1012) — array elements.

Print single positive integer m in the first line — the number of elements in desired sequence. In the second line print m distinct integers — the sequence of indices of given array elements, which should be taken into the desired sequence.  If there are more than one such subsequence (e.g. subsequence of minimum possible number of elements and with minimum possible sum of elements), you can print any of them. If there are no such subsequences, print  - 1 in the only line.


Dr. Bruce Banner hates his enemies (like others don't). As we all know, he can barely talk when he turns into the incredible Hulk. That's why he asked you to help him to express his feelings.Hulk likes the Inception so much, and like that his feelings are complicated. They have n layers. The first layer is hate, second one is love, third one is hate and so on...For example if n = 1, then his feeling is "I hate it" or if n = 2 it's "I hate that I love it", and if n = 3 it's "I hate that I love that I hate it" and so on.Please help Dr. Banner.

The only line of the input contains a single integer n (1 ≤ n ≤ 100) — the number of layers of love and hate.

I hate it
I hate that I love it
I hate that I love that I hate it


Tony Stark is playing a game with his suits (they have auto-pilot now). He lives in Malibu. Malibu has n junctions numbered from 1 to n, connected with n - 1 roads. One can get from a junction to any other junction using these roads (graph of Malibu forms a tree).Tony has m suits. There's a special plan for each suit. The i-th suit will appear at the moment of time ti in the junction vi, and will move to junction ui using the shortest path between vi and ui with the speed ci roads per second (passing a junctions takes no time), and vanishing immediately when arriving at ui (if it reaches ui in time q, it's available there at moment q, but not in further moments). Also, suits move continuously (for example if vi ≠ ui, at time  it's in the middle of a road. Please note that if vi = ui it means the suit will be at junction number vi only at moment ti and then it vanishes. An explosion happens if at any moment of time two suits share the same exact location (it may be in a junction or somewhere on a road; while appearing, vanishing or moving).Your task is to tell Tony the moment of the the first explosion (if there will be any).

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100 000) — the number of junctions and the number of suits respectively. The next n - 1 lines contain the roads descriptions. Each line contains two integers ai and bi — endpoints of the i-th road (1 ≤ ai, bi ≤ n, ai ≠ bi). The next m lines contain the suit descriptions. The i-th of them contains four integers ti, ci, vi and ui (0 ≤ ti ≤ 10 000, 1 ≤ ci ≤ 10 000, 1 ≤ vi, ui ≤ n), meaning the i-th suit will appear at moment of time ti at the junction vi and will move to the junction ui with a speed ci roads per second.

If there would be no explosions at all, print -1 in the first and only line of output. Otherwise print the moment of the first explosion. Your answer will be considered correct if its relative or absolute error doesn't exceed 10 - 6.


Vasiliy lives at point (a, b) of the coordinate plane. He is hurrying up to work so he wants to get out of his house as soon as possible. New app suggested n available Beru-taxi nearby. The i-th taxi is located at point (xi, yi) and moves with a speed vi. Consider that each of n drivers will move directly to Vasiliy and with a maximum possible speed. Compute the minimum time when Vasiliy will get in any of Beru-taxi cars.

The first line of the input contains two integers a and b ( - 100 ≤ a, b ≤ 100) — coordinates of Vasiliy's home. The second line contains a single integer n (1 ≤ n ≤ 1000) — the number of available Beru-taxi cars nearby.  The i-th of the following n lines contains three integers xi, yi and vi ( - 100 ≤ xi, yi ≤ 100, 1 ≤ vi ≤ 100) — the coordinates of the i-th car and its speed. It's allowed that several cars are located at the same point. Also, cars may be located at exactly the same point where Vasiliy lives.

Print a single real value — the minimum time Vasiliy needs to get in any of the Beru-taxi cars. 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 .

NoteIn the first sample, first taxi will get to Vasiliy in time 2, and second will do this in time 1, therefore 1 is the answer.In the second sample, cars 2 and 3 will arrive simultaneously.


Peter Parker wants to play a game with Dr. Octopus. The game is about cycles. Cycle is a sequence of vertices, such that first one is connected with the second, second is connected with third and so on, while the last one is connected with the first one again. Cycle may consist of a single isolated vertex.Initially there are k cycles, i-th of them consisting of exactly vi vertices. Players play alternatively. Peter goes first. On each turn a player must choose a cycle with at least 2 vertices (for example, x vertices) among all available cycles and replace it by two cycles with p and x - p vertices where 1 ≤ p < x is chosen by the player. The player who cannot make a move loses the game (and his life!).Peter wants to test some configurations of initial cycle sets before he actually plays with Dr. Octopus. Initially he has an empty set. In the i-th test he adds a cycle with ai vertices to the set (this is actually a multiset because it can contain two or more identical cycles). After each test, Peter wants to know that if the players begin the game with the current set of cycles, who wins? Peter is pretty good at math, but now he asks you to help.

The first line of the input contains a single integer n (1 ≤ n ≤ 100 000) — the number of tests Peter is about to make. The second line contains n space separated integers a1, a2, ..., an (1 ≤ ai ≤ 109), i-th of them stands for the number of vertices in the cycle added before the i-th test.

Print the result of all tests in order they are performed. Print 1 if the player who moves first wins or 2 otherwise.

NoteIn the first sample test:In Peter's first test, there's only one cycle with 1 vertex. First player cannot make a move and loses.In his second test, there's one cycle with 1 vertex and one with 2. No one can make a move on the cycle with 1 vertex. First player can replace the second cycle with two cycles of 1 vertex and second player can't make any move and loses.In his third test, cycles have 1, 2 and 3 vertices. Like last test, no one can make a move on the first cycle. First player can replace the third cycle with one cycle with size 1 and one with size 2. Now cycles have 1, 1, 2, 2 vertices. Second player's only move is to replace a cycle of size 2 with 2 cycles of size 1. And cycles are 1, 1, 1, 1, 2. First player replaces the last cycle with 2 cycles with size 1 and wins.In the second sample test:Having cycles of size 1 is like not having them (because no one can make a move on them). In Peter's third test: There a cycle of size 5 (others don't matter). First player has two options: replace it with cycles of sizes 1 and 4 or 2 and 3.  If he replaces it with cycles of sizes 1 and 4: Only second cycle matters. Second player will replace it with 2 cycles of sizes 2. First player's only option to replace one of them with two cycles of size 1. Second player does the same thing with the other cycle. First player can't make any move and loses.  If he replaces it with cycles of sizes 2 and 3: Second player will replace the cycle of size 3 with two of sizes 1 and 2. Now only cycles with more than one vertex are two cycles of size 2. As shown in previous case, with 2 cycles of size 2 second player wins. So, either way first player loses.


Author has gone out of the stories about Vasiliy, so here is just a formal task description.You are given q queries and a multiset A, initially containing only integer 0. There are three types of queries:  "+ x" — add integer x to multiset A. "- x" — erase one occurrence of integer x from multiset A. It's guaranteed that at least one x is present in the multiset A before this query. "? x" — you are given integer x and need to compute the value , i.e. the maximum value of bitwise exclusive OR (also know as XOR) of integer x and some integer y from the multiset A.Multiset is a set, where equal elements are allowed.

The first line of the input contains a single integer q (1 ≤ q ≤ 200 000) — the number of queries Vasiliy has to perform. Each of the following q lines of the input contains one of three characters '+', '-' or '?' and an integer xi (1 ≤ xi ≤ 109). It's guaranteed that there is at least one query of the third type. Note, that the integer 0 will always be present in the set A.

For each query of the type '?' print one integer — the maximum value of bitwise exclusive OR (XOR) of integer xi and some integer from the multiset A.

NoteAfter first five operations multiset A contains integers 0, 8, 9, 11, 6 and 1.The answer for the sixth query is integer  — maximum among integers , , ,  and .


Vasiliy likes to rest after a hard work, so you may often meet him in some bar nearby. As all programmers do, he loves the famous drink "Beecola", which can be bought in n different shops in the city. It's known that the price of one bottle in the shop i is equal to xi coins.Vasiliy plans to buy his favorite drink for q consecutive days. He knows, that on the i-th day he will be able to spent mi coins. Now, for each of the days he want to know in how many different shops he can buy a bottle of "Beecola".

The first line of the input contains a single integer n (1 ≤ n ≤ 100 000) — the number of shops in the city that sell Vasiliy's favourite drink. The second line contains n integers xi (1 ≤ xi ≤ 100 000) — prices of the bottles of the drink in the i-th shop. The third line contains a single integer q (1 ≤ q ≤ 100 000) — the number of days Vasiliy plans to buy the drink. Then follow q lines each containing one integer mi (1 ≤ mi ≤ 109) — the number of coins Vasiliy can spent on the i-th day.

Print q integers. The i-th of them should be equal to the number of shops where Vasiliy will be able to buy a bottle of the drink on the i-th day.

NoteOn the first day, Vasiliy won't be able to buy a drink in any of the shops.On the second day, Vasiliy can buy a drink in the shops 1, 2, 3 and 4.On the third day, Vasiliy can buy a drink only in the shop number 1.Finally, on the last day Vasiliy can buy a drink in any shop.


Vasiliy finally got to work, where there is a huge amount of tasks waiting for him. Vasiliy is given a matrix consisting of n rows and m columns and q tasks. Each task is to swap two submatrices of the given matrix.For each task Vasiliy knows six integers ai, bi, ci, di, hi, wi, where ai is the index of the row where the top-left corner of the first rectangle is located, bi is the index of its column, ci is the index of the row of the top-left corner of the second rectangle, di is the index of its column, hi is the height of the rectangle and wi is its width.It's guaranteed that two rectangles in one query do not overlap and do not touch, that is, no cell belongs to both rectangles, and no two cells belonging to different rectangles share a side. However, rectangles are allowed to share an angle.Vasiliy wants to know how the matrix will look like after all tasks are performed.

The first line of the input contains three integers n, m and q (2 ≤ n, m ≤ 1000, 1 ≤ q ≤ 10 000) — the number of rows and columns in matrix, and the number of tasks Vasiliy has to perform. Then follow n lines containing m integers vi, j (1 ≤ vi, j ≤ 109) each — initial values of the cells of the matrix. Each of the following q lines contains six integers ai, bi, ci, di, hi, wi (1 ≤ ai, ci, hi ≤ n, 1 ≤ bi, di, wi ≤ m).

Print n lines containing m integers each — the resulting matrix.


Vasiliy is fond of solving different tasks. Today he found one he wasn't able to solve himself, so he asks you to help.Vasiliy is given n strings consisting of lowercase English letters. He wants them to be sorted in lexicographical order (as in the dictionary), but he is not allowed to swap any of them. The only operation he is allowed to do is to reverse any of them (first character becomes last, second becomes one before last and so on).To reverse the i-th string Vasiliy has to spent ci units of energy. He is interested in the minimum amount of energy he has to spent in order to have strings sorted in lexicographical order.String A is lexicographically smaller than string B if it is shorter than B (|A| < |B|) and is its prefix, or if none of them is a prefix of the other and at the first position where they differ character in A is smaller than the character in B.For the purpose of this problem, two equal strings nearby do not break the condition of sequence being sorted lexicographically.

The first line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of strings. The second line contains n integers ci (0 ≤ ci ≤ 109), the i-th of them is equal to the amount of energy Vasiliy has to spent in order to reverse the i-th string.  Then follow n lines, each containing a string consisting of lowercase English letters. The total length of these strings doesn't exceed 100 000.

If it is impossible to reverse some of the strings such that they will be located in lexicographical order, print  - 1. Otherwise, print the minimum total amount of energy Vasiliy has to spent.

NoteIn the second sample one has to reverse string 2 or string 3. To amount of energy required to reverse the string 3 is smaller.In the third sample, both strings do not change after reverse and they go in the wrong order, so the answer is  - 1.In the fourth sample, both strings consists of characters 'a' only, but in the sorted order string "aa" should go before string "aaa", thus the answer is  - 1.


Small, but very brave, mouse Brain was not accepted to summer school of young villains. He was upset and decided to postpone his plans of taking over the world, but to become a photographer instead.As you may know, the coolest photos are on the film (because you can specify the hashtag #film for such).Brain took a lot of colourful pictures on colored and black-and-white film. Then he developed and translated it into a digital form. But now, color and black-and-white photos are in one folder, and to sort them, one needs to spend more than one hour!As soon as Brain is a photographer not programmer now, he asks you to help him determine for a single photo whether it is colored or black-and-white.Photo can be represented as a matrix sized n × m, and each element of the matrix stores a symbol indicating corresponding pixel color. There are only 6 colors:   'C' (cyan) 'M' (magenta) 'Y' (yellow) 'W' (white) 'G' (grey) 'B' (black) The photo is considered black-and-white if it has only white, black and grey pixels in it. If there are any of cyan, magenta or yellow pixels in the photo then it is considered colored.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100) — the number of photo pixel matrix rows and columns respectively. Then n lines describing matrix rows follow. Each of them contains m space-separated characters describing colors of pixels in a row. Each character in the line is one of the 'C', 'M', 'Y', 'W', 'G' or 'B'.

Print the "#Black&White" (without quotes), if the photo is black-and-white and "#Color" (without quotes), if it is colored, in the only line.


Masha wants to open her own bakery and bake muffins in one of the n cities numbered from 1 to n. There are m bidirectional roads, each of whose connects some pair of cities.To bake muffins in her bakery, Masha needs to establish flour supply from some storage. There are only k storages, located in different cities numbered a1, a2, ..., ak.Unforunately the law of the country Masha lives in prohibits opening bakery in any of the cities which has storage located in it. She can open it only in one of another n - k cities, and, of course, flour delivery should be paid — for every kilometer of path between storage and bakery Masha should pay 1 ruble.Formally, Masha will pay x roubles, if she will open the bakery in some city b (ai ≠ b for every 1 ≤ i ≤ k) and choose a storage in some city s (s = aj for some 1 ≤ j ≤ k) and b and s are connected by some path of roads of summary length x (if there are more than one path, Masha is able to choose which of them should be used).Masha is very thrifty and rational. She is interested in a city, where she can open her bakery (and choose one of k storages and one of the paths between city with bakery and city with storage) and pay minimum possible amount of rubles for flour delivery. Please help Masha find this amount.

The first line of the input contains three integers n, m and k (1 ≤ n, m ≤ 105, 0 ≤ k ≤ n) — the number of cities in country Masha lives in, the number of roads between them and the number of flour storages respectively. Then m lines follow. Each of them contains three integers u, v and l (1 ≤ u, v ≤ n, 1 ≤ l ≤ 109, u ≠ v) meaning that there is a road between cities u and v of length of l kilometers . If k > 0, then the last line of the input contains k distinct integers a1, a2, ..., ak (1 ≤ ai ≤ n) — the number of cities having flour storage located in. If k = 0 then this line is not presented in the input.

Print the minimum possible amount of rubles Masha should pay for flour delivery in the only line. If the bakery can not be opened (while satisfying conditions) in any of the n cities, print  - 1 in the only line.

NoteImage illustrates the first sample case. Cities with storage located in and the road representing the answer are darkened.


Katya studies in a fifth grade. Recently her class studied right triangles and the Pythagorean theorem. It appeared, that there are triples of positive integers such that you can construct a right triangle with segments of lengths corresponding to triple. Such triples are called Pythagorean triples.For example, triples (3, 4, 5), (5, 12, 13) and (6, 8, 10) are Pythagorean triples.Here Katya wondered if she can specify the length of some side of right triangle and find any Pythagorean triple corresponding to such length? Note that the side which length is specified can be a cathetus as well as hypotenuse.Katya had no problems with completing this task. Will you do the same?

The only line of the input contains single integer n (1 ≤ n ≤ 109) — the length of some side of a right triangle.

Print two integers m and k (1 ≤ m, k ≤ 1018), such that n, m and k form a Pythagorean triple, in the only line. In case if there is no any Pythagorean triple containing integer n, print  - 1 in the only line. If there are many answers, print any of them.


Like all children, Alesha loves New Year celebration. During the celebration he and his whole family dress up the fir-tree. Like all children, Alesha likes to play with garlands — chains consisting of a lightbulbs.Alesha uses a grid field sized n × m for playing. The rows of the field are numbered from 1 to n from the top to the bottom and columns are numbered from 1 to m from the left to the right.Alesha has k garlands which he places at the field. He does so in the way such that each lightbulb of each garland lies in the center of some cell in the field, and each cell contains at most one lightbulb. Of course lightbulbs, which are neighbours in some garland, appears in cells neighbouring by a side.The example of garland placing.Each garland is turned off or turned on at any moment. If some garland is turned on then each of its lightbulbs is turned on, the same applies for garland turned off. Each lightbulb in the whole garland set is unique, and thus, being turned on, brings Alesha some pleasure, described by an integer value. Turned off lightbulbs don't bring Alesha any pleasure.Alesha can turn garlands on and off and wants to know the sum of pleasure value which the lightbulbs, placed in the centers of the cells in some rectangular part of the field, bring him. Initially all the garlands are turned on.Alesha is still very little and can't add big numbers. He extremely asks you to help him.

The first line of the input contains three integers n, m and k (1 ≤ n, m, k ≤ 2000) — the number of field rows, the number of field columns and the number of garlands placed at the field respectively. Next lines contains garlands set description in the following format: The first line of a single garland description contains a single integer len (1 ≤ len ≤ 2000) — the number of lightbulbs in the garland. Each of the next len lines contains three integers i, j and w (1 ≤ i ≤ n, 1 ≤ j ≤ m, 1 ≤ w ≤ 109) — the coordinates of the cell containing a lightbullb and pleasure value Alesha gets from it if it is turned on. The lightbulbs are given in the order they are forming a chain in the garland. It is guaranteed that neighbouring lightbulbs are placed in the cells neighbouring by a side. The next line contains single integer q (1 ≤ q ≤ 106) — the number of events in Alesha's game. The next q lines describes events in chronological order. The i-th of them describes the i-th event in the one of the following formats:   SWITCH i — Alesha turns off i-th garland if it is turned on, or turns it on if it is turned off. It is guaranteed that 1 ≤ i ≤ k.  ASK x1 y1 x2 y2 — Alesha wants to know the sum of pleasure values the lightbulbs, placed in a rectangular part of the field. Top-left cell of a part has coordinates (x1, y1) and right-bottom cell has coordinates (x2, y2). It is guaranteed that 1 ≤ x1 ≤ x2 ≤ n and 1 ≤ y1 ≤ y2 ≤ m. There is no more than 2000 events of this type in the input.  All the numbers in the input are integers. Please note that the input is quite large, so be careful while using some input ways. In particular, it's not recommended to use cin in codes on C++ and class Scanner in codes on Java.

For each ASK operation print the sum Alesha wants to know in a separate line. Print the answers in chronological order.

NoteThis image illustrates the first sample case.


You are given a non-empty string s consisting of lowercase English letters. You have to pick exactly one non-empty substring of s and shift all its letters 'z'  'y'  'x'  'b'  'a'  'z'. In other words, each character is replaced with the previous character of English alphabet and 'a' is replaced with 'z'.What is the lexicographically minimum string that can be obtained from s by performing this shift exactly once?

The only line of the input contains the string s (1 ≤ |s| ≤ 100 000) consisting of lowercase English letters.

Print the lexicographically minimum string that can be obtained from s by shifting letters of exactly one non-empty substring.

NoteString s is lexicographically smaller than some other string t of the same length if there exists some 1 ≤ i ≤ |s|, such that s1 = t1, s2 = t2, ..., si - 1 = ti - 1, and si < ti.


For each string s consisting of characters '0' and '1' one can define four integers a00, a01, a10 and a11, where axy is the number of subsequences of length 2 of the string s equal to the sequence {x, y}. In these problem you are given four integers a00, a01, a10, a11 and have to find any non-empty string s that matches them, or determine that there is no such string. One can prove that if at least one answer exists, there exists an answer of length no more than 1 000 000.

The only line of the input contains four non-negative integers a00, a01, a10 and a11. Each of them doesn't exceed 109.

If there exists a non-empty string that matches four integers from the input, print it in the only line of the output. Otherwise, print "Impossible". The length of your answer must not exceed 1 000 000.


Recently in school Alina has learned what are the persistent data structures: they are data structures that always preserves the previous version of itself and access to it when it is modified.After reaching home Alina decided to invent her own persistent data structure. Inventing didn't take long: there is a bookcase right behind her bed. Alina thinks that the bookcase is a good choice for a persistent data structure. Initially the bookcase is empty, thus there is no book at any position at any shelf.The bookcase consists of n shelves, and each shelf has exactly m positions for books at it. Alina enumerates shelves by integers from 1 to n and positions at shelves — from 1 to m. Initially the bookcase is empty, thus there is no book at any position at any shelf in it.Alina wrote down q operations, which will be consecutively applied to the bookcase. Each of the operations has one of four types: 1 i j — Place a book at position j at shelf i if there is no book at it. 2 i j — Remove the book from position j at shelf i if there is a book at it. 3 i — Invert book placing at shelf i. This means that from every position at shelf i which has a book at it, the book should be removed, and at every position at shelf i which has not book at it, a book should be placed. 4 k — Return the books in the bookcase in a state they were after applying k-th operation. In particular, k = 0 means that the bookcase should be in initial state, thus every book in the bookcase should be removed from its position.After applying each of operation Alina is interested in the number of books in the bookcase. Alina got 'A' in the school and had no problem finding this values. Will you do so?

The first line of the input contains three integers n, m and q (1 ≤ n, m ≤ 103, 1 ≤ q ≤ 105) — the bookcase dimensions and the number of operations respectively. The next q lines describes operations in chronological order — i-th of them describes i-th operation in one of the four formats described in the statement. It is guaranteed that shelf indices and position indices are correct, and in each of fourth-type operation the number k corresponds to some operation before it or equals to 0.

For each operation, print the number of books in the bookcase after applying it in a separate line. The answers should be printed in chronological order.

NoteThis image illustrates the second sample case.


Tree is a connected acyclic graph. Suppose you are given a tree consisting of n vertices. The vertex of this tree is called centroid if the size of each connected component that appears if this vertex is removed from the tree doesn't exceed .You are given a tree of size n and can perform no more than one edge replacement. Edge replacement is the operation of removing one edge from the tree (without deleting incident vertices) and inserting one new edge (without adding new vertices) in such a way that the graph remains a tree. For each vertex you have to determine if it's possible to make it centroid by performing no more than one edge replacement.

The first line of the input contains an integer n (2 ≤ n ≤ 400 000) — the number of vertices in the tree. Each of the next n - 1 lines contains a pair of vertex indices ui and vi (1 ≤ ui, vi ≤ n) — endpoints of the corresponding edge.

Print n integers. The i-th of them should be equal to 1 if the i-th vertex can be made centroid by replacing no more than one edge, and should be equal to 0 otherwise.

NoteIn the first sample each vertex can be made a centroid. For example, in order to turn vertex 1 to centroid one have to replace the edge (2, 3) with the edge (1, 3).


At the entrance examination for the magistracy of the MSU Cyber-Mechanics Department Sasha got the question about Ford-Fulkerson algorithm. He knew the topic perfectly as he worked with it many times on programming competition. As the task for the question he was given a network with partially build flow that he had to use in order to demonstrate the workflow of the algorithm. He quickly finished to write the text and took a look at the problem only to understand that the given network is incorrect!Suppose you are given a directed graph G(V, E) with two special nodes s and t called source and sink. We denote as n the number of nodes in the graph, i.e. n = |V| and m stands for the number of directed edges in the graph, i.e. m = |E|. For the purpose of this problem we always consider node 1 to be the source and node n to be the sink. In addition, for each edge of the graph e we define the capacity function c(e) and flow function f(e). Function f(e) represents the correct flow if the following conditions are satisfied:  For each edge  the flow is non-negative and does not exceed capacity c(e), i.e. 0 ≤ f(e) ≤ c(e).  For each node , that is not source or sink (v ≠ s and v ≠ t) the sum of flows of all edges going in v is equal to the sum of the flows among all edges going out from v. In other words, there is no flow stuck in v. It was clear that as the exam was prepared last night and there are plenty of mistakes in the tasks. Sasha asked one of the professors to fix the network or give the correct task, but the reply was that the magistrate student should be able to fix the network himself. As the professor doesn't want the task to become easier, he asks Sasha to fix the network in a such way that the total number of changes is minimum possible. Sasha is not allowed to remove edges, add new ones or reverse the direction of existing edges. The only thing he is able to do is to change capacity function c(e) and flow function f(e). Moreover, all the values should remain non-negative integers. There is no requirement on the flow to be maximum in any sense.Find the minimum possible total change of the functions f(e) and c(e) that Sasha has to make in order to make the flow correct. The total change is defined as the sum of absolute differences, i.e. if new functions are f * (e) and c * (e), then the total change is .

The first line of the input contains two integers n and m (2 ≤ n ≤ 100, 0 ≤ m ≤ 100) — the number of nodes and edges in the graph respectively. Each of the following m lines contains the description of the edges, consisting of four integers ui, vi, ci and fi (1 ≤ ui, vi ≤ n, ui ≠ vi, 0 ≤ ci, fi ≤ 1 000 000) — index of the node the edges starts from, the index of the node the edge goes to, current capacity and flow value. Node number 1 is the source, and node number n is the sink. It's guaranteed that no edge goes to the source, and no edges starts in the sink. Given graph contains no self-loops but may contain multiple edges.

Print one integer — the minimum total sum of changes that Sasha has to do in order to get the correct flow description.

NoteIn the first sample, the flow is initially correct. Note, that the flow is not maximum, but this is not required.In the second sample, the flow value of the only edge is greater than its capacity. There are two ways to fix this: either increase the capacity up to 2 or reduce the flow down to 1.In the third sample, there is only 1 unit of flow coming to vertex 2, but there are 2 units going out of it. One of the possible solutions is to reduce the value of the flow on the second edge by 1.In the fourth sample, there is isolated circulation of flow, but this description is correct by definition.


Kolya is going to make fresh orange juice. He has n oranges of sizes a1, a2, ..., an. Kolya will put them in the juicer in the fixed order, starting with orange of size a1, then orange of size a2 and so on. To be put in the juicer the orange must have size not exceeding b, so if Kolya sees an orange that is strictly greater he throws it away and continues with the next one.The juicer has a special section to collect waste. It overflows if Kolya squeezes oranges of the total size strictly greater than d. When it happens Kolya empties the waste section (even if there are no more oranges) and continues to squeeze the juice. How many times will he have to empty the waste section?

The first line of the input contains three integers n, b and d (1 ≤ n ≤ 100 000, 1 ≤ b ≤ d ≤ 1 000 000) — the number of oranges, the maximum size of the orange that fits in the juicer and the value d, which determines the condition when the waste section should be emptied. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 1 000 000) — sizes of the oranges listed in the order Kolya is going to try to put them in the juicer.

Print one integer — the number of times Kolya will have to empty the waste section.

NoteIn the first sample, Kolya will squeeze the juice from two oranges and empty the waste section afterwards.In the second sample, the orange won't fit in the juicer so Kolya will have no juice at all.


Vasya takes part in the orienteering competition. There are n checkpoints located along the line at coordinates x1, x2, ..., xn. Vasya starts at the point with coordinate a. His goal is to visit at least n - 1 checkpoint in order to finish the competition. Participant are allowed to visit checkpoints in arbitrary order.Vasya wants to pick such checkpoints and the order of visiting them that the total distance travelled is minimized. He asks you to calculate this minimum possible value.

The first line of the input contains two integers n and a (1 ≤ n ≤ 100 000,  - 1 000 000 ≤ a ≤ 1 000 000) — the number of checkpoints and Vasya's starting position respectively. The second line contains n integers x1, x2, ..., xn ( - 1 000 000 ≤ xi ≤ 1 000 000) — coordinates of the checkpoints.

Print one integer — the minimum distance Vasya has to travel in order to visit at least n - 1 checkpoint.

NoteIn the first sample Vasya has to visit at least two checkpoints. The optimal way to achieve this is the walk to the third checkpoints (distance is 12 - 10 = 2) and then proceed to the second one (distance is 12 - 7 = 5). The total distance is equal to 2 + 5 = 7.In the second sample it's enough to visit only one checkpoint so Vasya should just walk to the point  - 10.


You are given n points on a line with their coordinates xi. Find the point x so the sum of distances to the given points is minimal.

The first line contains integer n (1 ≤ n ≤ 3·105) — the number of points on the line. The second line contains n integers xi ( - 109 ≤ xi ≤ 109) — the coordinates of the given n points.

Print the only integer x — the position of the optimal point on the line. If there are several optimal points print the position of the leftmost one. It is guaranteed that the answer is always the integer.


Find an n × n matrix with different numbers from 1 to n2, so the sum in each row, column and both main diagonals are odd.

The only line contains odd integer n (1 ≤ n ≤ 49).

Print n lines with n integers. All the integers should be different and from 1 to n2. The sum in each row, column and both main diagonals should be odd.


The only king stands on the standard chess board. You are given his position in format "cd", where c is the column from 'a' to 'h' and d is the row from '1' to '8'. Find the number of moves permitted for the king.Check the king's moves here https://en.wikipedia.org/wiki/King_(chess).  King moves from the position e4

The only line contains the king's position in the format "cd", where 'c' is the column from 'a' to 'h' and 'd' is the row from '1' to '8'.

Print the only integer x — the number of moves permitted for the king.


You are given two arithmetic progressions: a1k + b1 and a2l + b2. Find the number of integers x such that L ≤ x ≤ R and x = a1k' + b1 = a2l' + b2, for some integers k', l' ≥ 0.

The only line contains six integers a1, b1, a2, b2, L, R (0 < a1, a2 ≤ 2·109,  - 2·109 ≤ b1, b2, L, R ≤ 2·109, L ≤ R).


zscoder wants to generate an input file for some programming competition problem.His input is a string consisting of n letters 'a'. He is too lazy to write a generator so he will manually generate the input in a text editor.Initially, the text editor is empty. It takes him x seconds to insert or delete a letter 'a' from the text file and y seconds to copy the contents of the entire text file, and duplicate it.zscoder wants to find the minimum amount of time needed for him to create the input file of exactly n letters 'a'. Help him to determine the amount of time needed to generate the input.

The only line contains three integers n, x and y (1 ≤ n ≤ 107, 1 ≤ x, y ≤ 109) — the number of letters 'a' in the input file and the parameters from the problem statement.

Print the only integer t — the minimum amount of time needed to generate the input file.


Alex studied well and won the trip to student camp Alushta, located on the seashore. Unfortunately, it's the period of the strong winds now and there is a chance the camp will be destroyed! Camp building can be represented as the rectangle of n + 2 concrete blocks height and m blocks width.Every day there is a breeze blowing from the sea. Each block, except for the blocks of the upper and lower levers, such that there is no block to the left of it is destroyed with the probability . Similarly, each night the breeze blows in the direction to the sea. Thus, each block (again, except for the blocks of the upper and lower levers) such that there is no block to the right of it is destroyed with the same probability p. Note, that blocks of the upper and lower level are indestructible, so there are only n·m blocks that can be destroyed.The period of the strong winds will last for k days and k nights. If during this period the building will split in at least two connected components, it will collapse and Alex will have to find another place to spend summer.Find the probability that Alex won't have to look for other opportunities and will be able to spend the summer in this camp.

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 1500) that define the size of the destructible part of building. The second line of the input contains two integers a and b (1 ≤ a ≤ b ≤ 109) that define the probability p. It's guaranteed that integers a and b are coprime.  The third line contains a single integer k (0 ≤ k ≤ 100 000) — the number of days and nights strong wind will blow for.

Consider the answer as an irreducible fraction is equal to . Print one integer equal to . It's guaranteed that within the given constraints .

NoteIn the first sample, each of the four blocks is destroyed with the probability . There are 7 scenarios that result in building not collapsing, and the probability we are looking for is equal to , so you should print


You should process m queries over a set D of strings. Each query is one of three kinds:  Add a string s to the set D. It is guaranteed that the string s was not added before.  Delete a string s from the set D. It is guaranteed that the string s is in the set D.  For the given string s find the number of occurrences of the strings from the set D. If some string p from D has several occurrences in s you should count all of them. Note that you should solve the problem in online mode. It means that you can't read the whole input at once. You can read each query only after writing the answer for the last query of the third type. Use functions fflush in C++ and BufferedWriter.flush in Java languages after each writing in your program.

The first line contains integer m (1 ≤ m ≤ 3·105) — the number of queries. Each of the next m lines contains integer t (1 ≤ t ≤ 3) and nonempty string s — the kind of the query and the string to process. All strings consist of only lowercase English letters. The sum of lengths of all strings in the input will not exceed 3·105.

For each query of the third kind print the only integer c — the desired number of occurrences in the string s.

10
1 abc
1 bcd
1 abcd
3 abcd
2 abcd
3 abcd
2 bcd
3 abcd
2 abc
3 abcd


ZS the Coder and Chris the Baboon are travelling to Udayland! To get there, they have to get on the special IOI bus. The IOI bus has n rows of seats. There are 4 seats in each row, and the seats are separated into pairs by a walkway. When ZS and Chris came, some places in the bus was already occupied.ZS and Chris are good friends. They insist to get a pair of neighbouring empty seats. Two seats are considered neighbouring if they are in the same row and in the same pair. Given the configuration of the bus, can you help ZS and Chris determine where they should sit?

The first line of the input contains a single integer n (1 ≤ n ≤ 1000) — the number of rows of seats in the bus. Then, n lines follow. Each line contains exactly 5 characters, the first two of them denote the first pair of seats in the row, the third character denotes the walkway (it always equals '|') and the last two of them denote the second pair of seats in the row.  Each character, except the walkway, equals to 'O' or to 'X'. 'O' denotes an empty seat, 'X' denotes an occupied seat. See the sample cases for more details.

If it is possible for Chris and ZS to sit at neighbouring empty seats, print "YES" (without quotes) in the first line. In the next n lines print the bus configuration, where the characters in the pair of seats for Chris and ZS is changed with characters '+'. Thus the configuration should differ from the input one by exactly two charaters (they should be equal to 'O' in the input and to '+' in the output). If there is no pair of seats for Chris and ZS, print "NO" (without quotes) in a single line. If there are multiple solutions, you may print any of them.

NoteNote that the following is an incorrect configuration for the first sample case because the seats must be in the same pair.O+|+XXO|XXOX|OOXX|OXOO|OOOO|XX


ZS the Coder and Chris the Baboon has arrived at Udayland! They walked in the park where n trees grow. They decided to be naughty and color the trees in the park. The trees are numbered with integers from 1 to n from left to right.Initially, tree i has color ci. ZS the Coder and Chris the Baboon recognizes only m different colors, so 0 ≤ ci ≤ m, where ci = 0 means that tree i is uncolored.ZS the Coder and Chris the Baboon decides to color only the uncolored trees, i.e. the trees with ci = 0. They can color each of them them in any of the m colors from 1 to m. Coloring the i-th tree with color j requires exactly pi, j litres of paint.The two friends define the beauty of a coloring of the trees as the minimum number of contiguous groups (each group contains some subsegment of trees) you can split all the n trees into so that each group contains trees of the same color. For example, if the colors of the trees from left to right are 2, 1, 1, 1, 3, 2, 2, 3, 1, 3, the beauty of the coloring is 7, since we can partition the trees into 7 contiguous groups of the same color : {2}, {1, 1, 1}, {3}, {2, 2}, {3}, {1}, {3}. ZS the Coder and Chris the Baboon wants to color all uncolored trees so that the beauty of the coloring is exactly k. They need your help to determine the minimum amount of paint (in litres) needed to finish the job.Please note that the friends can't color the trees that are already colored.

The first line contains three integers, n, m and k (1 ≤ k ≤ n ≤ 100, 1 ≤ m ≤ 100) — the number of trees, number of colors and beauty of the resulting coloring respectively. The second line contains n integers c1, c2, ..., cn (0 ≤ ci ≤ m), the initial colors of the trees. ci equals to 0 if the tree number i is uncolored, otherwise the i-th tree has color ci. Then n lines follow. Each of them contains m integers. The j-th number on the i-th of them line denotes pi, j (1 ≤ pi, j ≤ 109) — the amount of litres the friends need to color i-th tree with color j. pi, j's are specified even for the initially colored trees, but such trees still can't be colored.

Print a single integer, the minimum amount of paint needed to color the trees. If there are no valid tree colorings of beauty k, print  - 1.

NoteIn the first sample case, coloring the trees with colors 2, 1, 1 minimizes the amount of paint used, which equals to 2 + 3 + 5 = 10. Note that 1, 1, 1 would not be valid because the beauty of such coloring equals to 1 ({1, 1, 1} is a way to group the trees into a single group of the same color).In the second sample case, all the trees are colored, but the beauty of the coloring is 3, so there is no valid coloring, and the answer is  - 1.In the last sample case, all the trees are colored and the beauty of the coloring matches k, so no paint is used and the answer is 0.


ZS the Coder and Chris the Baboon has explored Udayland for quite some time. They realize that it consists of n towns numbered from 1 to n. There are n directed roads in the Udayland. i-th of them goes from town i to some other town ai (ai ≠ i). ZS the Coder can flip the direction of any road in Udayland, i.e. if it goes from town A to town B before the flip, it will go from town B to town A after.ZS the Coder considers the roads in the Udayland confusing, if there is a sequence of distinct towns A1, A2, ..., Ak (k > 1) such that for every 1 ≤ i < k there is a road from town Ai to town Ai + 1 and another road from town Ak to town A1. In other words, the roads are confusing if some of them form a directed cycle of some towns.Now ZS the Coder wonders how many sets of roads (there are 2n variants) in initial configuration can he choose to flip such that after flipping each road in the set exactly once, the resulting network will not be confusing.Note that it is allowed that after the flipping there are more than one directed road from some town and possibly some towns with no roads leading out of it, or multiple roads between any pair of cities.

The first line of the input contains single integer n (2 ≤ n ≤ 2·105) — the number of towns in Udayland. The next line contains n integers a1, a2, ..., an (1 ≤ ai ≤ n, ai ≠ i), ai denotes a road going from town i to town ai.

Print a single integer — the number of ways to flip some set of the roads so that the resulting whole set of all roads is not confusing. Since this number may be too large, print the answer modulo 109 + 7.

NoteConsider the first sample case. There are 3 towns and 3 roads. The towns are numbered from 1 to 3 and the roads are , ,  initially. Number the roads 1 to 3 in this order. The sets of roads that ZS the Coder can flip (to make them not confusing) are {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}. Note that the empty set is invalid because if no roads are flipped, then towns 1, 2, 3 is form a directed cycle, so it is confusing. Similarly, flipping all roads is confusing too. Thus, there are a total of 6 possible sets ZS the Coder can flip.The sample image shows all possible ways of orienting the roads from the first sample such that the network is not confusing.


ZS the Coder and Chris the Baboon arrived at the entrance of Udayland. There is a n × n magic grid on the entrance which is filled with integers. Chris noticed that exactly one of the cells in the grid is empty, and to enter Udayland, they need to fill a positive integer into the empty cell.Chris tried filling in random numbers but it didn't work. ZS the Coder realizes that they need to fill in a positive integer such that the numbers in the grid form a magic square. This means that he has to fill in a positive integer so that the sum of the numbers in each row of the grid (), each column of the grid (), and the two long diagonals of the grid (the main diagonal —  and the secondary diagonal — ) are equal. Chris doesn't know what number to fill in. Can you help Chris find the correct positive integer to fill in or determine that it is impossible?

The first line of the input contains a single integer n (1 ≤ n ≤ 500) — the number of rows and columns of the magic grid. n lines follow, each of them contains n integers. The j-th number in the i-th of them denotes ai, j (1 ≤ ai, j ≤ 109 or ai, j = 0), the number in the i-th row and j-th column of the magic grid. If the corresponding cell is empty, ai, j will be equal to 0. Otherwise, ai, j is positive. It is guaranteed that there is exactly one pair of integers i, j (1 ≤ i, j ≤ n) such that ai, j = 0.

Output a single integer, the positive integer x (1 ≤ x ≤ 1018) that should be filled in the empty cell so that the whole grid becomes a magic square. If such positive integer x does not exist, output  - 1 instead. If there are multiple solutions, you may print any of them.

NoteIn the first sample case, we can fill in 9 into the empty cell to make the resulting grid a magic square. Indeed, The sum of numbers in each row is:4 + 9 + 2 = 3 + 5 + 7 = 8 + 1 + 6 = 15.The sum of numbers in each column is:4 + 3 + 8 = 9 + 5 + 1 = 2 + 7 + 6 = 15.The sum of numbers in the two diagonals is:4 + 5 + 6 = 2 + 5 + 8 = 15.In the third sample case, it is impossible to fill a number in the empty square such that the resulting grid is a magic square.


There are n integers b1, b2, ..., bn written in a row. For all i from 1 to n, values ai are defined by the crows performing the following procedure:  The crow sets ai initially 0.  The crow then adds bi to ai, subtracts bi + 1, adds the bi + 2 number, and so on until the n'th number. Thus, ai = bi - bi + 1 + bi + 2 - bi + 3.... Memory gives you the values a1, a2, ..., an, and he now wants you to find the initial numbers b1, b2, ..., bn written in the row? Can you do it?

The first line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of integers written in the row. The next line contains n, the i'th of which is ai ( - 109 ≤ ai ≤ 109) — the value of the i'th number.

Print n integers corresponding to the sequence b1, b2, ..., bn. It's guaranteed that the answer is unique and fits in 32-bit integer type.

NoteIn the first sample test, the crows report the numbers 6, - 4, 8, - 2, and 3 when he starts at indices 1, 2, 3, 4 and 5 respectively. It is easy to check that the sequence 2 4 6 1 3 satisfies the reports. For example, 6 = 2 - 4 + 6 - 1 + 3, and  - 4 = 4 - 6 + 1 - 3.In the second sample test, the sequence 1,  - 3, 4, 11, 6 satisfies the reports. For example, 5 = 11 - 6 and 6 = 6.


Memory is performing a walk on the two-dimensional plane, starting at the origin. He is given a string s with his directions for motion: An 'L' indicates he should move one unit left.  An 'R' indicates he should move one unit right.  A 'U' indicates he should move one unit up.  A 'D' indicates he should move one unit down.But now Memory wants to end at the origin. To do this, he has a special trident. This trident can replace any character in s with any of 'L', 'R', 'U', or 'D'. However, because he doesn't want to wear out the trident, he wants to make the minimum number of edits possible. Please tell Memory what is the minimum number of changes he needs to make to produce a string that, when walked, will end at the origin, or if there is no such string.

The first and only line contains the string s (1 ≤ |s| ≤ 100 000) — the instructions Memory is given.

If there is a string satisfying the conditions, output a single integer — the minimum number of edits required. In case it's not possible to change the sequence in such a way that it will bring Memory to to the origin, output -1.

NoteIn the first sample test, Memory is told to walk right, then right, then up. It is easy to see that it is impossible to edit these instructions to form a valid walk.In the second sample test, Memory is told to walk up, then down, then up, then right. One possible solution is to change s to "LDUR". This string uses 1 edit, which is the minimum possible. It also ends at the origin.


Memory is now interested in the de-evolution of objects, specifically triangles. He starts with an equilateral triangle of side length x, and he wishes to perform operations to obtain an equilateral triangle of side length y.In a single second, he can modify the length of a single side of the current triangle such that it remains a non-degenerate triangle (triangle of positive area). At any moment of time, the length of each side should be integer.What is the minimum number of seconds required for Memory to obtain the equilateral triangle of side length y?

The first and only line contains two integers x and y (3 ≤ y < x ≤ 100 000) — the starting and ending equilateral triangle side lengths respectively.

Print a single integer — the minimum number of seconds required for Memory to obtain the equilateral triangle of side length y if he starts with the equilateral triangle of side length x.

NoteIn the first sample test, Memory starts with an equilateral triangle of side length 6 and wants one of side length 3. Denote a triangle with sides a, b, and c as (a, b, c). Then, Memory can do .In the second sample test, Memory can do .In the third sample test, Memory can do: .


Memory and his friend Lexa are competing to get higher score in one popular computer game. Memory starts with score a and Lexa starts with score b. In a single turn, both Memory and Lexa get some integer in the range [ - k;k] (i.e. one integer among  - k,  - k + 1,  - k + 2, ...,  - 2,  - 1, 0, 1, 2, ..., k - 1, k) and add them to their current scores. The game has exactly t turns. Memory and Lexa, however, are not good at this game, so they both always get a random integer at their turn.Memory wonders how many possible games exist such that he ends with a strictly higher score than Lexa. Two games are considered to be different if in at least one turn at least one player gets different score. There are (2k + 1)2t games in total. Since the answer can be very large, you should print it modulo 109 + 7. Please solve this problem for Memory.

The first and only line of input contains the four integers a, b, k, and t (1 ≤ a, b ≤ 100, 1 ≤ k ≤ 1000, 1 ≤ t ≤ 100) — the amount Memory and Lexa start with, the number k, and the number of turns respectively.

Print the number of possible games satisfying the conditions modulo 1 000 000 007 (109 + 7) in one line.

NoteIn the first sample test, Memory starts with 1 and Lexa starts with 2. If Lexa picks  - 2, Memory can pick 0, 1, or 2 to win. If Lexa picks  - 1, Memory can pick 1 or 2 to win. If Lexa picks 0, Memory can pick 2 to win. If Lexa picks 1 or 2, Memory cannot win. Thus, there are 3 + 2 + 1 = 6 possible games in which Memory wins.


ZS the Coder has recently found an interesting concept called the Birthday Paradox. It states that given a random set of 23 people, there is around 50% chance that some two of them share the same birthday. ZS the Coder finds this very interesting, and decides to test this with the inhabitants of Udayland.In Udayland, there are 2n days in a year. ZS the Coder wants to interview k people from Udayland, each of them has birthday in one of 2n days (each day with equal probability). He is interested in the probability of at least two of them have the birthday at the same day. ZS the Coder knows that the answer can be written as an irreducible fraction . He wants to find the values of A and B (he does not like to deal with floating point numbers). Can you help him?

The first and only line of the input contains two integers n and k (1 ≤ n ≤ 1018, 2 ≤ k ≤ 1018), meaning that there are 2n days in a year and that ZS the Coder wants to interview exactly k people.

If the probability of at least two k people having the same birthday in 2n days long year equals  (A ≥ 0, B ≥ 1, ), print the A and B in a single line. Since these numbers may be too large, print them modulo 106 + 3. Note that A and B must be coprime before their remainders modulo 106 + 3 are taken.

NoteIn the first sample case, there are 23 = 8 days in Udayland. The probability that 2 people have the same birthday among 2 people is clearly , so A = 1, B = 8.In the second sample case, there are only 21 = 2 days in Udayland, but there are 3 people, so it is guaranteed that two of them have the same birthday. Thus, the probability is 1 and A = B = 1.


There are n casinos lined in a row. If Memory plays at casino i, he has probability pi to win and move to the casino on the right (i + 1) or exit the row (if i = n), and a probability 1 - pi to lose and move to the casino on the left (i - 1) or also exit the row (if i = 1). We say that Memory dominates on the interval i... j if he completes a walk such that,  He starts on casino i.  He never looses in casino i.  He finishes his walk by winning in casino j. Note that Memory can still walk left of the 1-st casino and right of the casino n and that always finishes the process.Now Memory has some requests, in one of the following forms:  1 i a b: Set .  2 l r: Print the probability that Memory will dominate on the interval l... r, i.e. compute the probability that Memory will first leave the segment l... r after winning at casino r, if she starts in casino l. It is guaranteed that at any moment of time p is a non-decreasing sequence, i.e. pi ≤ pi + 1 for all i from 1 to n - 1.Please help Memory by answering all his requests!

The first line of the input contains two integers n and q(1 ≤ n, q ≤ 100 000),  — number of casinos and number of requests respectively. The next n lines each contain integers ai and bi (1 ≤ ai < bi ≤ 109)  —  is the probability pi of winning in casino i. The next q lines each contain queries of one of the types specified above (1 ≤ a < b ≤ 109, 1 ≤ i ≤ n, 1 ≤ l ≤ r ≤ n). It's guaranteed that there will be at least one query of type 2, i.e. the output will be non-empty. Additionally, it is guaranteed that p forms a non-decreasing sequence at all times.

Print a real number for every request of type 2 — the probability that boy will "dominate" on that interval. Your answer will be considered correct if its absolute error does not exceed 10 - 4. Namely: let's assume that one of your answers is a, and the corresponding answer of the jury is b. The checker program will consider your answer correct if |a - b| ≤ 10 - 4.


Filya just learned new geometry object — rectangle. He is given a field consisting of n × n unit cells. Rows are numbered from bottom to top with integer from 1 to n. Columns are numbered from left to right with integers from 1 to n. Cell, located at the intersection of the row r and column c is denoted as (r, c). Filya has painted two rectangles, such that their sides are parallel to coordinate axes and each cell lies fully inside or fully outside each of them. Moreover, no cell lies in both rectangles.Later, hedgehog Filya became interested in the location of his rectangles but was unable to find the sheet of paper they were painted on. They were taken by Sonya and now she wants to play a little game with Filya. He tells her a query rectangle and she replies with the number of initial rectangles that lie fully inside the given query rectangle. The query rectangle should match the same conditions as initial rectangles. Rectangle lies fully inside the query if each o its cells lies inside the query.Filya knows Sonya really well, so is sure that if he asks more than 200 questions she will stop to reply.

The first line of the input contains an integer n (2 ≤ n ≤ 216) — size of the field. For each query an integer between 0 and 2 is returned — the number of initial rectangles that lie fully inside the query rectangle.

To make a query you have to print "? x1 y1 x2 y2" (without quotes) (1 ≤ x1 ≤ x2 ≤ n, 1 ≤ y1 ≤ y2 ≤ n), where (x1, y1) stands for the position of the bottom left cell of the query and (x2, y2) stands for the up right cell of the query. You are allowed to ask no more than 200 queries. After each query you should perform "flush" operation and read the answer. In case you suppose you've already determined the location of two rectangles (or run out of queries) you should print "! x11 y11 x12 y12 x21 y21 x22 y22" (without quotes), where first four integers describe the bottom left and up right cells of the first rectangle, and following four describe the corresponding cells of the second rectangle. You can print the rectangles in an arbitrary order. After you have printed the answer, print the end of the line and perform "flush". Your program should terminate immediately after it print the answer.


Sonya was unable to think of a story for this problem, so here comes the formal description.You are given the array containing n positive integers. At one turn you can pick any element and increase or decrease it by 1. The goal is the make the array strictly increasing by making the minimum possible number of operations. You are allowed to change elements in any way, they can become negative or equal to 0.

The first line of the input contains a single integer n (1 ≤ n ≤ 3000) — the length of the array. Next line contains n integer ai (1 ≤ ai ≤ 109).

Print the minimum number of operation required to make the array strictly increasing.

NoteIn the first sample, the array is going to look as follows:2 3 5 6 7 9 11|2 - 2| + |1 - 3| + |5 - 5| + |11 - 6| + |5 - 7| + |9 - 9| + |11 - 11| = 9And for the second sample:1 2 3 4 5|5 - 1| + |4 - 2| + |3 - 3| + |2 - 4| + |1 - 5| = 12