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Solve the programming task below in a Python markdown code block. The Happy Farm 5 creators decided to invent the mechanism of cow grazing. The cows in the game are very slow and they move very slowly, it can even be considered that they stand still. However, carnivores should always be chased off them. For that a young player Vasya decided to make the shepherd run round the cows along one and the same closed path. It is very important that the cows stayed strictly inside the area limited by the path, as otherwise some cows will sooner or later be eaten. To be absolutely sure in the cows' safety, Vasya wants the path completion time to be minimum. The new game is launched for different devices, including mobile phones. That's why the developers decided to quit using the arithmetics with the floating decimal point and use only the arithmetics of integers. The cows and the shepherd in the game are represented as points on the plane with integer coordinates. The playing time is modeled by the turns. During every turn the shepherd can either stay where he stands or step in one of eight directions: horizontally, vertically, or diagonally. As the coordinates should always remain integer, then the length of a horizontal and vertical step is equal to 1, and the length of a diagonal step is equal to <image>. The cows do not move. You have to minimize the number of moves the shepherd needs to run round the whole herd. Input The first line contains an integer N which represents the number of cows in the herd (1 ≤ N ≤ 105). Each of the next N lines contains two integers Xi and Yi which represent the coordinates of one cow of (\|Xi\|, \|Yi\| ≤ 106). Several cows can stand on one point. Output Print the single number — the minimum number of moves in the sought path. Examples Input 4 1 1 5 1 5 3 1 3 Output 16 Note Picture for the example test: The coordinate grid is painted grey, the coordinates axes are painted black, the cows are painted red and the sought route is painted green. <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In Berland a bus travels along the main street of the capital. The street begins from the main square and looks like a very long segment. There are n bus stops located along the street, the i-th of them is located at the distance ai from the central square, all distances are distinct, the stops are numbered in the order of increasing distance from the square, that is, ai < ai + 1 for all i from 1 to n - 1. The bus starts its journey from the first stop, it passes stops 2, 3 and so on. It reaches the stop number n, turns around and goes in the opposite direction to stop 1, passing all the intermediate stops in the reverse order. After that, it again starts to move towards stop n. During the day, the bus runs non-stop on this route. The bus is equipped with the Berland local positioning system. When the bus passes a stop, the system notes down its number. One of the key features of the system is that it can respond to the queries about the distance covered by the bus for the parts of its path between some pair of stops. A special module of the system takes the input with the information about a set of stops on a segment of the path, a stop number occurs in the set as many times as the bus drove past it. This module returns the length of the traveled segment of the path (or -1 if it is impossible to determine the length uniquely). The operation of the module is complicated by the fact that stop numbers occur in the request not in the order they were visited but in the non-decreasing order. For example, if the number of stops is 6, and the part of the bus path starts at the bus stop number 5, ends at the stop number 3 and passes the stops as follows: <image>, then the request about this segment of the path will have form: 3, 4, 5, 5, 6. If the bus on the segment of the path from stop 5 to stop 3 has time to drive past the 1-th stop (i.e., if we consider a segment that ends with the second visit to stop 3 on the way from 5), then the request will have form: 1, 2, 2, 3, 3, 4, 5, 5, 6. You will have to repeat the Berland programmers achievement and implement this function. Input The first line contains integer n (2 ≤ n ≤ 2·105) — the number of stops. The second line contains n integers (1 ≤ ai ≤ 109) — the distance from the i-th stop to the central square. The numbers in the second line go in the increasing order. The third line contains integer m (1 ≤ m ≤ 4·105) — the number of stops the bus visited on some segment of the path. The fourth line contains m integers (1 ≤ bi ≤ n) — the sorted list of numbers of the stops visited by the bus on the segment of the path. The number of a stop occurs as many times as it was visited by a bus. It is guaranteed that the query corresponds to some segment of the path. Output In the single line please print the distance covered by a bus. If it is impossible to determine it unambiguously, print - 1. Examples Input 6 2 3 5 7 11 13 5 3 4 5 5 6 Output 10 Input 6 2 3 5 7 11 13 9 1 2 2 3 3 4 5 5 6 Output 16 Input 3 10 200 300 4 1 2 2 3 Output -1 Input 3 1 2 3 4 1 2 2 3 Output 3 Note The first test from the statement demonstrates the first example shown in the statement of the problem. The second test from the statement demonstrates the second example shown in the statement of the problem. In the third sample there are two possible paths that have distinct lengths, consequently, the sought length of the segment isn't defined uniquely. In the fourth sample, even though two distinct paths correspond to the query, they have the same lengths, so the sought length of the segment is defined uniquely. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In the spirit of the holidays, Saitama has given Genos two grid paths of length n (a weird gift even by Saitama's standards). A grid path is an ordered sequence of neighbouring squares in an infinite grid. Two squares are neighbouring if they share a side. One example of a grid path is (0, 0) → (0, 1) → (0, 2) → (1, 2) → (1, 1) → (0, 1) → ( - 1, 1). Note that squares in this sequence might be repeated, i.e. path has self intersections. Movement within a grid path is restricted to adjacent squares within the sequence. That is, from the i-th square, one can only move to the (i - 1)-th or (i + 1)-th squares of this path. Note that there is only a single valid move from the first and last squares of a grid path. Also note, that even if there is some j-th square of the path that coincides with the i-th square, only moves to (i - 1)-th and (i + 1)-th squares are available. For example, from the second square in the above sequence, one can only move to either the first or third squares. To ensure that movement is not ambiguous, the two grid paths will not have an alternating sequence of three squares. For example, a contiguous subsequence (0, 0) → (0, 1) → (0, 0) cannot occur in a valid grid path. One marble is placed on the first square of each grid path. Genos wants to get both marbles to the last square of each grid path. However, there is a catch. Whenever he moves one marble, the other marble will copy its movement if possible. For instance, if one marble moves east, then the other marble will try and move east as well. By try, we mean if moving east is a valid move, then the marble will move east. Moving north increases the second coordinate by 1, while moving south decreases it by 1. Similarly, moving east increases first coordinate by 1, while moving west decreases it. Given these two valid grid paths, Genos wants to know if it is possible to move both marbles to the ends of their respective paths. That is, if it is possible to move the marbles such that both marbles rest on the last square of their respective paths. Input The first line of the input contains a single integer n (2 ≤ n ≤ 1 000 000) — the length of the paths. The second line of the input contains a string consisting of n - 1 characters (each of which is either 'N', 'E', 'S', or 'W') — the first grid path. The characters can be thought of as the sequence of moves needed to traverse the grid path. For example, the example path in the problem statement can be expressed by the string "NNESWW". The third line of the input contains a string of n - 1 characters (each of which is either 'N', 'E', 'S', or 'W') — the second grid path. Output Print "YES" (without quotes) if it is possible for both marbles to be at the end position at the same time. Print "NO" (without quotes) otherwise. In both cases, the answer is case-insensitive. Examples Input 7 NNESWW SWSWSW Output YES Input 3 NN SS Output NO Note In the first sample, the first grid path is the one described in the statement. Moreover, the following sequence of moves will get both marbles to the end: NNESWWSWSW. In the second sample, no sequence of moves can get both marbles to the end. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Famil Door wants to celebrate his birthday with his friends from Far Far Away. He has n friends and each of them can come to the party in a specific range of days of the year from ai to bi. Of course, Famil Door wants to have as many friends celebrating together with him as possible. Far cars are as weird as Far Far Away citizens, so they can only carry two people of opposite gender, that is exactly one male and one female. However, Far is so far from here that no other transportation may be used to get to the party. Famil Door should select some day of the year and invite some of his friends, such that they all are available at this moment and the number of male friends invited is equal to the number of female friends invited. Find the maximum number of friends that may present at the party. Input The first line of the input contains a single integer n (1 ≤ n ≤ 5000) — then number of Famil Door's friends. Then follow n lines, that describe the friends. Each line starts with a capital letter 'F' for female friends and with a capital letter 'M' for male friends. Then follow two integers ai and bi (1 ≤ ai ≤ bi ≤ 366), providing that the i-th friend can come to the party from day ai to day bi inclusive. Output Print the maximum number of people that may come to Famil Door's party. Examples Input 4 M 151 307 F 343 352 F 117 145 M 24 128 Output 2 Input 6 M 128 130 F 128 131 F 131 140 F 131 141 M 131 200 M 140 200 Output 4 Note In the first sample, friends 3 and 4 can come on any day in range [117, 128]. In the second sample, friends with indices 3, 4, 5 and 6 can come on day 140. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The life goes up and down, just like nice sequences. Sequence t1, t2, ..., tn is called nice if the following two conditions are satisfied: * ti < ti + 1 for each odd i < n; * ti > ti + 1 for each even i < n. For example, sequences (2, 8), (1, 5, 1) and (2, 5, 1, 100, 99, 120) are nice, while (1, 1), (1, 2, 3) and (2, 5, 3, 2) are not. Bear Limak has a sequence of positive integers t1, t2, ..., tn. This sequence is not nice now and Limak wants to fix it by a single swap. He is going to choose two indices i < j and swap elements ti and tj in order to get a nice sequence. Count the number of ways to do so. Two ways are considered different if indices of elements chosen for a swap are different. Input The first line of the input contains one integer n (2 ≤ n ≤ 150 000) — the length of the sequence. The second line contains n integers t1, t2, ..., tn (1 ≤ ti ≤ 150 000) — the initial sequence. It's guaranteed that the given sequence is not nice. Output Print the number of ways to swap two elements exactly once in order to get a nice sequence. Examples Input 5 2 8 4 7 7 Output 2 Input 4 200 150 100 50 Output 1 Input 10 3 2 1 4 1 4 1 4 1 4 Output 8 Input 9 1 2 3 4 5 6 7 8 9 Output 0 Note In the first sample, there are two ways to get a nice sequence with one swap: 1. Swap t2 = 8 with t4 = 7. 2. Swap t1 = 2 with t5 = 7. In the second sample, there is only one way — Limak should swap t1 = 200 with t4 = 50. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is an interactive problem. In the output section below you will see the information about flushing the output. Bear Limak thinks of some hidden number — an integer from interval [2, 100]. Your task is to say if the hidden number is prime or composite. Integer x > 1 is called prime if it has exactly two distinct divisors, 1 and x. If integer x > 1 is not prime, it's called composite. You can ask up to 20 queries about divisors of the hidden number. In each query you should print an integer from interval [2, 100]. The system will answer "yes" if your integer is a divisor of the hidden number. Otherwise, the answer will be "no". For example, if the hidden number is 14 then the system will answer "yes" only if you print 2, 7 or 14. When you are done asking queries, print "prime" or "composite" and terminate your program. You will get the Wrong Answer verdict if you ask more than 20 queries, or if you print an integer not from the range [2, 100]. Also, you will get the Wrong Answer verdict if the printed answer isn't correct. You will get the Idleness Limit Exceeded verdict if you don't print anything (but you should) or if you forget about flushing the output (more info below). Input After each query you should read one string from the input. It will be "yes" if the printed integer is a divisor of the hidden number, and "no" otherwise. Output Up to 20 times you can ask a query — print an integer from interval [2, 100] in one line. You have to both print the end-of-line character and flush the output. After flushing you should read a response from the input. In any moment you can print the answer "prime" or "composite" (without the quotes). After that, flush the output and terminate your program. To flush you can use (just after printing an integer and end-of-line): * fflush(stdout) in C++; * System.out.flush() in Java; * stdout.flush() in Python; * flush(output) in Pascal; * See the documentation for other languages. Hacking. To hack someone, as the input you should print the hidden number — one integer from the interval [2, 100]. Of course, his/her solution won't be able to read the hidden number from the input. Examples Input yes no yes Output 2 80 5 composite Input no yes no no no Output 58 59 78 78 2 prime Note The hidden number in the first query is 30. In a table below you can see a better form of the provided example of the communication process. <image> The hidden number is divisible by both 2 and 5. Thus, it must be composite. Note that it isn't necessary to know the exact value of the hidden number. In this test, the hidden number is 30. <image> 59 is a divisor of the hidden number. In the interval [2, 100] there is only one number with this divisor. The hidden number must be 59, which is prime. Note that the answer is known even after the second query and you could print it then and terminate. Though, it isn't forbidden to ask unnecessary queries (unless you exceed the limit of 20 queries). The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. 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. Input 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. Output 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. Examples Input 3 3 1 1 3 1 2 2 Output 4 2 0 Input 5 2 1 5 5 1 Output 16 9 Input 100000 1 300 400 Output 9999800001 Note On 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. <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Polycarp is a music editor at the radio station. He received a playlist for tomorrow, that can be represented as a sequence a1, a2, ..., an, where ai is a band, which performs the i-th song. Polycarp likes bands with the numbers from 1 to m, but he doesn't really like others. We define as bj the number of songs the group j is going to perform tomorrow. Polycarp wants to change the playlist in such a way that the minimum among the numbers b1, b2, ..., bm will be as large as possible. Find this maximum possible value of the minimum among the bj (1 ≤ j ≤ m), and the minimum number of changes in the playlist Polycarp needs to make to achieve it. One change in the playlist is a replacement of the performer of the i-th song with any other group. Input The first line of the input contains two integers n and m (1 ≤ m ≤ n ≤ 2000). The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109), where ai is the performer of the i-th song. Output In the first line print two integers: the maximum possible value of the minimum among the bj (1 ≤ j ≤ m), where bj is the number of songs in the changed playlist performed by the j-th band, and the minimum number of changes in the playlist Polycarp needs to make. In the second line print the changed playlist. If there are multiple answers, print any of them. Examples Input 4 2 1 2 3 2 Output 2 1 1 2 1 2 Input 7 3 1 3 2 2 2 2 1 Output 2 1 1 3 3 2 2 2 1 Input 4 4 1000000000 100 7 1000000000 Output 1 4 1 2 3 4 Note In the first sample, after Polycarp's changes the first band performs two songs (b1 = 2), and the second band also performs two songs (b2 = 2). Thus, the minimum of these values equals to 2. It is impossible to achieve a higher minimum value by any changes in the playlist. In the second sample, after Polycarp's changes the first band performs two songs (b1 = 2), the second band performs three songs (b2 = 3), and the third band also performs two songs (b3 = 2). Thus, the best minimum value is 2. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Hongcow likes solving puzzles. One day, Hongcow finds two identical puzzle pieces, with the instructions "make a rectangle" next to them. The pieces can be described by an n by m grid of characters, where the character 'X' denotes a part of the puzzle and '.' denotes an empty part of the grid. It is guaranteed that the puzzle pieces are one 4-connected piece. See the input format and samples for the exact details on how a jigsaw piece will be specified. The puzzle pieces are very heavy, so Hongcow cannot rotate or flip the puzzle pieces. However, he is allowed to move them in any directions. The puzzle pieces also cannot overlap. You are given as input the description of one of the pieces. Determine if it is possible to make a rectangle from two identical copies of the given input. The rectangle should be solid, i.e. there should be no empty holes inside it or on its border. Keep in mind that Hongcow is not allowed to flip or rotate pieces and they cannot overlap, i.e. no two 'X' from different pieces can share the same position. Input The first line of input will contain two integers n and m (1 ≤ n, m ≤ 500), the dimensions of the puzzle piece. The next n lines will describe the jigsaw piece. Each line will have length m and will consist of characters '.' and 'X' only. 'X' corresponds to a part of the puzzle piece, '.' is an empty space. It is guaranteed there is at least one 'X' character in the input and that the 'X' characters form a 4-connected region. Output Output "YES" if it is possible for Hongcow to make a rectangle. Output "NO" otherwise. Examples Input 2 3 XXX XXX Output YES Input 2 2 .X XX Output NO Input 5 5 ..... ..X.. ..... ..... ..... Output YES Note For the first sample, one example of a rectangle we can form is as follows 111222 111222 For the second sample, it is impossible to put two of those pieces without rotating or flipping to form a rectangle. In the third sample, we can shift the first tile by one to the right, and then compose the following rectangle: ..... ..XX. ..... ..... ..... The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Jon Snow is on the lookout for some orbs required to defeat the white walkers. There are k different types of orbs and he needs at least one of each. One orb spawns daily at the base of a Weirwood tree north of the wall. The probability of this orb being of any kind is equal. As the north of wall is full of dangers, he wants to know the minimum number of days he should wait before sending a ranger to collect the orbs such that the probability of him getting at least one of each kind of orb is at least <image>, where ε < 10 - 7. To better prepare himself, he wants to know the answer for q different values of pi. Since he is busy designing the battle strategy with Sam, he asks you for your help. Input First line consists of two space separated integers k, q (1 ≤ k, q ≤ 1000) — number of different kinds of orbs and number of queries respectively. Each of the next q lines contain a single integer pi (1 ≤ pi ≤ 1000) — i-th query. Output Output q lines. On i-th of them output single integer — answer for i-th query. Examples Input 1 1 1 Output 1 Input 2 2 1 2 Output 2 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A positive integer number n is written on a blackboard. It consists of not more than 105 digits. You have to transform it into a beautiful number by erasing some of the digits, and you want to erase as few digits as possible. The number is called beautiful if it consists of at least one digit, doesn't have leading zeroes and is a multiple of 3. For example, 0, 99, 10110 are beautiful numbers, and 00, 03, 122 are not. Write a program which for the given n will find a beautiful number such that n can be transformed into this number by erasing as few digits as possible. You can erase an arbitraty set of digits. For example, they don't have to go one after another in the number n. If it's impossible to obtain a beautiful number, print -1. If there are multiple answers, print any of them. Input The first line of input contains n — a positive integer number without leading zeroes (1 ≤ n < 10100000). Output Print one number — any beautiful number obtained by erasing as few as possible digits. If there is no answer, print - 1. Examples Input 1033 Output 33 Input 10 Output 0 Input 11 Output -1 Note In the first example it is enough to erase only the first digit to obtain a multiple of 3. But if we erase the first digit, then we obtain a number with a leading zero. So the minimum number of digits to be erased is two. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Sagheer is playing a game with his best friend Soliman. He brought a tree with n nodes numbered from 1 to n and rooted at node 1. The i-th node has ai apples. This tree has a special property: the lengths of all paths from the root to any leaf have the same parity (i.e. all paths have even length or all paths have odd length). Sagheer and Soliman will take turns to play. Soliman will make the first move. The player who can't make a move loses. In each move, the current player will pick a single node, take a non-empty subset of apples from it and do one of the following two things: 1. eat the apples, if the node is a leaf. 2. move the apples to one of the children, if the node is non-leaf. Before Soliman comes to start playing, Sagheer will make exactly one change to the tree. He will pick two different nodes u and v and swap the apples of u with the apples of v. Can you help Sagheer count the number of ways to make the swap (i.e. to choose u and v) after which he will win the game if both players play optimally? (u, v) and (v, u) are considered to be the same pair. Input The first line will contain one integer n (2 ≤ n ≤ 105) — the number of nodes in the apple tree. The second line will contain n integers a1, a2, ..., an (1 ≤ ai ≤ 107) — the number of apples on each node of the tree. The third line will contain n - 1 integers p2, p3, ..., pn (1 ≤ pi ≤ n) — the parent of each node of the tree. Node i has parent pi (for 2 ≤ i ≤ n). Node 1 is the root of the tree. It is guaranteed that the input describes a valid tree, and the lengths of all paths from the root to any leaf will have the same parity. Output On a single line, print the number of different pairs of nodes (u, v), u ≠ v such that if they start playing after swapping the apples of both nodes, Sagheer will win the game. (u, v) and (v, u) are considered to be the same pair. Examples Input 3 2 2 3 1 1 Output 1 Input 3 1 2 3 1 1 Output 0 Input 8 7 2 2 5 4 3 1 1 1 1 1 4 4 5 6 Output 4 Note In the first sample, Sagheer can only win if he swapped node 1 with node 3. In this case, both leaves will have 2 apples. If Soliman makes a move in a leaf node, Sagheer can make the same move in the other leaf. If Soliman moved some apples from a root to a leaf, Sagheer will eat those moved apples. Eventually, Soliman will not find a move. In the second sample, There is no swap that will make Sagheer win the game. Note that Sagheer must make the swap even if he can win with the initial tree. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Your security guard friend recently got a new job at a new security company. The company requires him to patrol an area of the city encompassing exactly N city blocks, but they let him choose which blocks. That is, your friend must walk the perimeter of a region whose area is exactly N blocks. Your friend is quite lazy and would like your help to find the shortest possible route that meets the requirements. The city is laid out in a square grid pattern, and is large enough that for the sake of the problem it can be considered infinite. Input Input will consist of a single integer N (1 ≤ N ≤ 106), the number of city blocks that must be enclosed by the route. Output Print the minimum perimeter that can be achieved. Examples Input 4 Output 8 Input 11 Output 14 Input 22 Output 20 Note Here are some possible shapes for the examples: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recently Luba bought a very interesting book. She knows that it will take t seconds to read the book. Luba wants to finish reading as fast as she can. But she has some work to do in each of n next days. The number of seconds that Luba has to spend working during i-th day is ai. If some free time remains, she can spend it on reading. Help Luba to determine the minimum number of day when she finishes reading. It is guaranteed that the answer doesn't exceed n. Remember that there are 86400 seconds in a day. Input The first line contains two integers n and t (1 ≤ n ≤ 100, 1 ≤ t ≤ 106) — the number of days and the time required to read the book. The second line contains n integers ai (0 ≤ ai ≤ 86400) — the time Luba has to spend on her work during i-th day. Output Print the minimum day Luba can finish reading the book. It is guaranteed that answer doesn't exceed n. Examples Input 2 2 86400 86398 Output 2 Input 2 86400 0 86400 Output 1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Carol is currently curling. She has n disks each with radius r on the 2D plane. Initially she has all these disks above the line y = 10100. She then will slide the disks towards the line y = 0 one by one in order from 1 to n. When she slides the i-th disk, she will place its center at the point (xi, 10100). She will then push it so the disk’s y coordinate continuously decreases, and x coordinate stays constant. The disk stops once it touches the line y = 0 or it touches any previous disk. Note that once a disk stops moving, it will not move again, even if hit by another disk. Compute the y-coordinates of centers of all the disks after all disks have been pushed. Input The first line will contain two integers n and r (1 ≤ n, r ≤ 1 000), the number of disks, and the radius of the disks, respectively. The next line will contain n integers x1, x2, ..., xn (1 ≤ xi ≤ 1 000) — the x-coordinates of the disks. Output Print a single line with n numbers. The i-th number denotes the y-coordinate of the center of the i-th disk. The output will be accepted if it has absolute or relative error at most 10 - 6. Namely, let's assume that your answer for a particular value of a coordinate is a and the answer of the jury is b. The checker program will consider your answer correct if <image> for all coordinates. Example Input 6 2 5 5 6 8 3 12 Output 2 6.0 9.87298334621 13.3370849613 12.5187346573 13.3370849613 Note The final positions of the disks will look as follows: <image> In particular, note the position of the last disk. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Little walrus Fangy loves math very much. That's why when he is bored he plays with a number performing some operations. Fangy takes some positive integer x and wants to get a number one from it. While x is not equal to 1, Fangy repeats the following action: if x is odd, then he adds 1 to it, otherwise he divides x by 2. Fangy knows that for any positive integer number the process ends in finite time. How many actions should Fangy perform to get a number one from number x? Input The first line contains a positive integer x in a binary system. It is guaranteed that the first digit of x is different from a zero and the number of its digits does not exceed 106. Output Print the required number of actions. Examples Input 1 Output 0 Input 1001001 Output 12 Input 101110 Output 8 Note Let's consider the third sample. Number 101110 is even, which means that we should divide it by 2. After the dividing Fangy gets an odd number 10111 and adds one to it. Number 11000 can be divided by 2 three times in a row and get number 11. All that's left is to increase the number by one (we get 100), and then divide it by 2 two times in a row. As a result, we get 1. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Consider a new order of english alphabets (a to z), that we are not aware of. What we have though, dictionary of words, ordered according to the new order. Our task is to give each alphabet a rank, ordered list of words. The rank of an alphabet is the minimum integer R that can be given to it, such that: if alphabet Ai has rank Ri and alphabet Aj has Rj, then Rj > Ri if and only if Aj > Ai Rj < Ri if and only if Aj < Ai Points to note: We only need to find the order of alphabets that are present in the words There might be some cases where a strict ordering of words cannot be inferred completely, given the limited information. In that case, give the least possible Rank to every alphabet It can be assumed that there will be no direct or indirect conflict in different inferences got from the input Input: First line consists of integer 'W', the number of words. 1 ≤ W ≤ 100 Second line onwards, there are 'W' lines, given ordered list of words. The words will be non-empty, and will be of length ≤ 20 characters. Also all the characters will be lowercase alphabets, a to z Output: The alphabets, in order of their Rank, for alphabets with the same rank, print them in the order as we know it today Examples 1) Input: 2 ba ab Output: b a Explanation: ab comes after ba implies that b < a 2) Input: 3 abc aca ca Ouput: ab c Explanation: from the first 2 words, we know b < c, and from the last 2 words, we know a < c. We don't have an ordering between a and b, so we give both of them the same rank, and print them in the order as we know today: ab SAMPLE INPUT 6 aaabc aaade bacd cde cda cca SAMPLE OUTPUT e a b d c The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Flip the world is a game. In this game a matrix of size N*M is given, which consists of numbers. Each number can be 1 or 0 only. The rows are numbered from 1 to N, and the columns are numbered from 1 to M. Following steps can be called as a single move. Select two integers x,y (1 ≤ x ≤ N\; and\; 1 ≤ y ≤ M) i.e. one square on the matrix. All the integers in the rectangle denoted by (1,1) and (x,y) i.e. rectangle having top-left and bottom-right points as (1,1) and (x,y) are toggled(1 is made 0 and 0 is made 1). For example, in this matrix (N=4 and M=3) 101 110 101 000 if we choose x=3 and y=2, the new state of matrix would be 011 000 011 000 For a given state of matrix, aim of the game is to reduce the matrix to a state where all numbers are 1. What is minimum number of moves required. INPUT: First line contains T, the number of testcases. Each testcase consists of two space-separated integers denoting N,M. Each of the next N lines contains string of size M denoting each row of the matrix. Each element is either 0 or 1. OUTPUT: For each testcase, print the minimum required moves. CONSTRAINTS: 1 ≤ T ≤ 30 1 ≤ N ≤ 20 1 ≤ M ≤ 20 SAMPLE INPUT 1 5 5 00011 00011 00011 11111 11111 SAMPLE OUTPUT 1 Explanation In one move we can choose 3,3 to make the whole matrix consisting of only 1s. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mike has a huge guitar collection which the whole band is jealous of. Brad, the lead guitarist of the band, gave Mike a challenge so that he can't use all his guitars. The band is on a world tour and Brad told Mike to assign numbers to all his guitars. After that, he told Mike that he can use the guitars in such a manner that no two concerts have the same sum when the numbers on the guitars used in that concert are added. Now, Mike has other important things to do, like performing sound checks and last minute rehersals, and so he asks for your help. Help him find how many concerts will he be able to perform in this World Tour with the restrictions imposed on him by Brad . Input: First line contains t, the number of test cases. Now, each test case consists of 2 lines, the first one contains N,the number of guitars that Mike has and the second line contains N space separated numbers assigned by Mike to his guitars denoted by a[i]. Output: For each test case you need to print a single integer denoting the number of concerts that Mike would be able to perform in this world tour. Constraints: 1 ≤ t ≤ 20 1 ≤ N ≤ 20 1 ≤ a[i] ≤ 100 Problem Setter: Rohit Mishra SAMPLE INPUT 3 1 1 2 1 2 3 1 2 3 SAMPLE OUTPUT 2 4 7 Explanation Consider the 3rd test case We have guitars numbered 1,2 and 3 So all the different possibilities are: {} - Sum=0 {1} - Sum=1 {2} - Sum=2 {3} - Sum=3 {1,2} - Sum=3 {1,3} - Sum=4 {2,3} - Sum=5 {1,2,3} - Sum=6 So, we see that 7 different sums are possible (3 is counted only once) and so output is 7. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given are integers A, B, and N. Find the maximum possible value of floor(Ax/B) - A × floor(x/B) for a non-negative integer x not greater than N. Here floor(t) denotes the greatest integer not greater than the real number t. Constraints * 1 ≤ A ≤ 10^{6} * 1 ≤ B ≤ 10^{12} * 1 ≤ N ≤ 10^{12} * All values in input are integers. Input Input is given from Standard Input in the following format: A B N Output Print the maximum possible value of floor(Ax/B) - A × floor(x/B) for a non-negative integer x not greater than N, as an integer. Examples Input 5 7 4 Output 2 Input 11 10 9 Output 9 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let us consider a grid of squares with N rows and N columns. You want to put some domino pieces on this grid. Each domino piece covers two squares that have a common side. Each square can be covered by at most one piece. For each row of the grid, let's define its quality as the number of domino pieces that cover at least one square in this row. We define the quality of each column similarly. Find a way to put at least one domino piece on the grid so that the quality of every row is equal to the quality of every column, or determine that such a placement doesn't exist. Constraints * 2 \le N \le 1000 Input Input is given from Standard Input in the following format: N Output If the required domino placement doesn't exist, print a single integer `-1`. Otherwise, output your placement as N strings of N characters each. If a square is not covered, the corresponding character must be `.` (a dot). Otherwise, it must contain a lowercase English letter. Squares covered by the same domino piece must contain the same letter. If two squares have a common side but belong to different pieces, they must contain different letters. Examples Input 6 Output aabb.. b..zz. ba.... .a..aa ..a..b ..a..b Input 2 Output -1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given are two strings s and t consisting of lowercase English letters. Determine if the number of non-negative integers i satisfying the following condition is finite, and find the maximum value of such i if the number is finite. * There exists a non-negative integer j such that the concatenation of i copies of t is a substring of the concatenation of j copies of s. Constraints * 1 \leq \|s\| \leq 5 \times 10^5 * 1 \leq \|t\| \leq 5 \times 10^5 * s and t consist of lowercase English letters. Input Input is given from Standard Input in the following format: s t Output If the number of non-negative integers i satisfying the following condition is finite, print the maximum value of such i; if the number is infinite, print `-1`. Examples Input abcabab ab Output 3 Input aa aaaaaaa Output -1 Input aba baaab Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Kurohashi has never participated in AtCoder Beginner Contest (ABC). The next ABC to be held is ABC N (the N-th ABC ever held). Kurohashi wants to make his debut in some ABC x such that all the digits of x in base ten are the same. What is the earliest ABC where Kurohashi can make his debut? Constraints * 100 \leq N \leq 999 * N is an integer. Input Input is given from Standard Input in the following format: N Output If the earliest ABC where Kurohashi can make his debut is ABC n, print n. Examples Input 111 Output 111 Input 112 Output 222 Input 750 Output 777 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In Finite Encyclopedia of Integer Sequences (FEIS), all integer sequences of lengths between 1 and N (inclusive) consisting of integers between 1 and K (inclusive) are listed. Let the total number of sequences listed in FEIS be X. Among those sequences, find the (X/2)-th (rounded up to the nearest integer) lexicographically smallest one. Constraints * 1 \leq N,K \leq 3 × 10^5 * N and K are integers. Input Input is given from Standard Input in the following format: K N Output Print the (X/2)-th (rounded up to the nearest integer) lexicographically smallest sequence listed in FEIS, with spaces in between, where X is the total number of sequences listed in FEIS. Examples Input 3 2 Output 2 1 Input 2 4 Output 1 2 2 2 Input 5 14 Output 3 3 3 3 3 3 3 3 3 3 3 3 2 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mr. Tanaka died leaving the orchard of HW Earl. The orchard is divided into H x W plots in the north, south, east, and west directions, and apples, oysters, and oranges are planted in each plot. Mr. Tanaka left such a will. Divide the orchard into as many relatives as possible on a parcel basis. However, if the same kind of fruit is planted in a plot that is located in either the north, south, east, or west direction of a plot, treat them as one large plot because the boundaries of the plot are unknown. For example, in the following 3 × 10 section ('li' represents an apple,'ka' represents an oyster, and'mi' represents a mandarin orange) <image> Eliminating the boundaries between plots with the same tree gives: <image> In the end, it will be divided into 10 compartments, or 10 people. The distribution must be completed before it snows and the boundaries of the plot disappear. Your job is to determine the number of plots to distribute based on the map of the orchard. Create a program that reads the map of the orchard and outputs the number of relatives who can receive the distribution. Input Given multiple datasets. Each dataset is given a string of H lines consisting of the characters H x W, starting with a line containing H, W (H, W ≤ 100) separated by blanks. Only three characters appear in this string:'@' for apples,'#' for oysters, and'' for oranges. The input ends with two lines of zeros. The number of datasets does not exceed 20. Output For each dataset, output the number of people to be distributed on one line. Examples Input 10 10 ####***@ @#@@@@##* @##**@@@ #***#@** ##@#@@## @@@@@@@# **#@@##* @@@@@##@ @#@##* @**#@@#@ 0 0 Output 33 Input 10 10 **@ @#@@@@##* @##**@@@ ***#@** @#@@## @@@@@@@# **#@@##* @@@@@##@ @#@##* @****#@@#@ 0 0 Output 33 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Welcome to PC Koshien, players. This year marks the 10th anniversary of Computer Koshien, but the number of questions and the total score will vary from year to year. Scores are set for each question according to the difficulty level. When the number of questions is 10 and the score of each question is given, create a program that outputs the total of them. input The input is given in the following format. s1 s2 .. .. s10 The input consists of 10 lines, and the i line is given the integer si (0 ≤ si ≤ 100) representing the score of problem i. output Output the total score on one line. Example Input 1 2 3 4 5 6 7 8 9 10 Output 55 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Dr .: Peter, do you know "Yes, I have a number"? Peter: I used to do it on TV the other day. You remember something by the number of characters in each word contained in a sentence. "Yes, I have a number", so it means "the number 3.14" and is a keyword for remembering pi. Dr .: Peter, that's not the case. This should be interpreted as 3.1416. The circumference ratio is 3.14159 ... That's why. Peter: Then, did you forcibly omit that program just because Japan teaches that the circumference ratio is 3.14? ?? Dr .: ... Let's just say that the pi taught in elementary school has finally returned from 3 to 3.14. Peter: Do you really remember this in the first place? Dr .: It may be difficult for Japanese people. It seems that English-speaking people use it, because it is difficult to make a ground ball in English. Peter: Even so, it seems to be a hassle to check. Dr .: Then, I want you to make a program that converts sentences into a sequence of the number of characters in a word. Peter: I understand. Please tell me the detailed specifications. Dr .: Let's enter one line of text. For simplicity, you can use sentences that contain only alphabets and blanks. For this sentence, output the length of the character string between the blank, the beginning of the sentence, and the end of the sentence in order. You can proceed if the number of characters in the string does not exceed 9. For example, Yes in "Yes I have" has 3 characters because it is separated by the beginning and the first space, and I is separated by the first and second spaces and has 1 character. Dr .: Then, don't forget the character string where the number of characters becomes 0 when there are consecutive blanks. Input Multiple datasets are given as input. For each dataset, a string containing alphabets and spaces is given on one line. When the character string is "END OF INPUT", it is the end of input. Do not output to this input. Output For each dataset, output a sequence of the number of characters for the string on one line. Example Input Yes I have a number How I wish I could calculate an unused color for space Thank you END OF INPUT Output 31416 31415926535 53 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Jim is planning to visit one of his best friends in a town in the mountain area. First, he leaves his hometown and goes to the destination town. This is called the go phase. Then, he comes back to his hometown. This is called the return phase. You are expected to write a program to find the minimum total cost of this trip, which is the sum of the costs of the go phase and the return phase. There is a network of towns including these two towns. Every road in this network is one-way, i.e., can only be used towards the specified direction. Each road requires a certain cost to travel. In addition to the cost of roads, it is necessary to pay a specified fee to go through each town on the way. However, since this is the visa fee for the town, it is not necessary to pay the fee on the second or later visit to the same town. The altitude (height) of each town is given. On the go phase, the use of descending roads is inhibited. That is, when going from town a to b, the altitude of a should not be greater than that of b. On the return phase, the use of ascending roads is inhibited in a similar manner. If the altitudes of a and b are equal, the road from a to b can be used on both phases. Input The input consists of multiple datasets, each in the following format. n m d2 e2 d3 e3 . . . dn-1 en-1 a1 b1 c1 a2 b2 c2 . . . am bm cm Every input item in a dataset is a non-negative integer. Input items in a line are separated by a space. n is the number of towns in the network. m is the number of (one-way) roads. You can assume the inequalities 2 ≤ n ≤ 50 and 0 ≤ m ≤ n(n−1) hold. Towns are numbered from 1 to n, inclusive. The town 1 is Jim's hometown, and the town n is the destination town. di is the visa fee of the town i, and ei is its altitude. You can assume 1 ≤ di ≤ 1000 and 1≤ei ≤ 999 for 2≤i≤n−1. The towns 1 and n do not impose visa fee. The altitude of the town 1 is 0, and that of the town n is 1000. Multiple towns may have the same altitude, but you can assume that there are no more than 10 towns with the same altitude. The j-th road is from the town aj to bj with the cost cj (1 ≤ j ≤ m). You can assume 1 ≤ aj ≤ n, 1 ≤ bj ≤ n, and 1 ≤ cj ≤ 1000. You can directly go from aj to bj, but not from bj to aj unless a road from bj to aj is separately given. There are no two roads connecting the same pair of towns towards the same direction, that is, for any i and j such that i ≠ j, ai ≠ aj or bi ≠ bj. There are no roads connecting a town to itself, that is, for any j, aj ≠ bj. The last dataset is followed by a line containing two zeros (separated by a space). Output For each dataset in the input, a line containing the minimum total cost, including the visa fees, of the trip should be output. If such a trip is not possible, output "-1". Example Input 3 6 3 1 1 2 1 2 3 1 3 2 1 2 1 1 1 3 4 3 1 4 3 6 5 1 1 2 1 2 3 1 3 2 1 2 1 1 1 3 4 3 1 4 4 5 3 1 3 1 1 2 5 2 3 5 3 4 5 4 2 5 3 1 5 2 1 2 1 1 0 0 Output 7 8 36 -1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Background The kindergarten attached to the University of Aizu is a kindergarten where children who love programming gather. Yu, one of the kindergarten children, loves darts as much as programming. Yu-kun was addicted to darts recently, but he got tired of ordinary darts, so he decided to make his own darts board. See darts for darts. Problem The contents of the darts that Yu-kun thought about are as follows. The infinitely wide darts board has several polygons with scores. The player has only one darts arrow. The player throws an arrow and stabs the arrow into one of the polygons to get the score written there. If you stab it in any other way, you will not get any points. Yu-kun decided where to throw the arrow, but it doesn't always stick exactly. The darts board is a two-dimensional plane, and the position Yu-kun is aiming for is a point (cx, cy). The place where the arrow thrown by Yu-kun sticks is selected with a uniform probability from any point included in the circle with radius r centered on the point (cx, cy). The coordinates of the exposed points do not have to be integers. Since the information on the darts board, the position (cx, cy) that Yu-kun aims at, and the radius r are given, answer the expected value of the score that Yu-kun can get. Constraints The input satisfies the following conditions. * All inputs are given as integers * 1 ≤ n ≤ 50 * 0 ≤ cx, cy, x, y ≤ 1000 * 1 ≤ r ≤ 100 * 3 ≤ p ≤ 10 * 1 ≤ score ≤ 100 * Polygon vertices are given in such an order that they visit adjacent vertices clockwise or counterclockwise. * The sides of a polygon do not have anything in common with the sides of another polygon * A polygon does not contain another polygon * If the arrow sticks on the side of the polygon, no score will be given. Input n cx cy r Information on the 0th polygon Information on the first polygon ... Information on the (n-1) th polygon n is the number of polygons on the darts board. Polygonal information is given in the following format. p score x0 y0 x1 y1 ... x (p-1) y (p-1) p represents the number of vertices of the polygon, and score represents the score written on the polygon. Each line segment of the polygon is a line segment connecting the vertices of (xi, yi) and (xi + 1, yi + 1) (i <p-1), and (xp-1, yp-1) and (x0, It is a line segment connecting the vertices of y0). Output Output the expected value of the score that Yu-kun can get in one line. Any number of digits after the decimal point may be output. However, the error in the answer must not exceed 0.000001 (10-6). Examples Input 1 2 2 1 4 1 0 0 2 0 2 2 0 2 Output 0.2500000000 Input 1 2 2 1 4 1 0 0 5 0 5 5 0 5 Output 1.0000000000 Input 4 3 3 2 3 1 1 1 3 3 1 5 4 2 2 0 5 0 4 2 3 2 3 3 4 3 6 1 6 5 4 4 3 4 4 4 5 6 2 6 Output 1.0574955319 Input 1 10 10 1 4 10 0 0 1 0 1 1 0 1 Output 0.0000000000 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In the International System of Units (SI), various physical quantities are expressed in the form of "numerical value + prefix + unit" using prefixes such as kilo, mega, and giga. For example, "3.5 kilometers", "5.1 milligrams", and so on. On the other hand, these physical quantities can be expressed as "3.5 * 10 ^ 3 meters" and "5.1 * 10 ^ -3 grams" using exponential notation. Measurement of physical quantities always includes errors. Therefore, there is a concept of significant figures to express how accurate the measured physical quantity is. In the notation considering significant figures, the last digit may contain an error, but the other digits are considered reliable. For example, if you write "1.23", the true value is 1.225 or more and less than 1.235, and the digits with one decimal place or more are reliable, but the second decimal place contains an error. The number of significant digits when a physical quantity is expressed in the form of "reliable digit + 1 digit including error" is called the number of significant digits. For example, "1.23" has 3 significant digits. If there is a 0 before the most significant non-zero digit, that 0 is not included in the number of significant digits. For example, "0.45" has two significant digits. If there is a 0 after the least significant non-zero digit, whether or not that 0 is included in the number of significant digits depends on the position of the decimal point. If there is a 0 to the right of the decimal point, that 0 is included in the number of significant digits. For example, "12.300" has 5 significant digits. On the other hand, when there is no 0 to the right of the decimal point like "12300", it is not clear whether to include the 0 on the right side in the number of significant digits, but it will be included in this problem. That is, the number of significant digits of "12300" is five. Natsume was given a physics problem as a school task. I have to do calculations related to significant figures and units, but the trouble is that I still don't understand how to use significant figures and units. Therefore, I would like you to create a program that automatically calculates them and help Natsume. What you write is a program that, given a prefixed notation, converts it to exponential notation with the same number of significant digits. The following 20 prefixes are used. * yotta = 10 ^ 24 * zetta = 10 ^ 21 * exa = 10 ^ 18 * peta = 10 ^ 15 * tera = 10 ^ 12 * giga = 10 ^ 9 * mega = 10 ^ 6 * kilo = 10 ^ 3 * hecto = 10 ^ 2 * deca = 10 ^ 1 * deci = 10 ^ -1 * centi = 10 ^ -2 * milli = 10 ^ -3 * micro = 10 ^ -6 * nano = 10 ^ -9 * pico = 10 ^ -12 * femto = 10 ^ -15 * ato = 10 ^ -18 * zepto = 10 ^ -21 * yocto = 10 ^ -24 Notes on Submission Multiple datasets are given in the above format. The first line of input data gives the number of datasets. Create a program that outputs the output for each data set in order in the above format. Input The input consists of only one line, which contains numbers, unit prefixes (if any), and units. Each is separated by a blank. In some cases, there is no unit prefix, in which case only numbers and units are included in the line. Units with the same name as the unit prefix do not appear. The most significant digit is never 0, except for the ones digit when a decimal is given. The number given is positive. The number given is 1000 digits or less including the decimal point, and the unit name is 50 characters or less. Output Output the quantity expressed in exponential notation in the form of a * 10 ^ b [unit]. However, 1 <= a <10. The difference between the singular and plural forms of the unit name is not considered. Output the unit name given to the input as it is. Example Input 7 12.3 kilo meters 0.45 mega watts 0.000000000000000000000001 yotta grams 1000000000000000000000000 yocto seconds 42 amperes 0.42 joules 1234.56789012345678901234567890 hecto pascals Output 1.23 * 10^4 meters 4.5 * 10^5 watts 1 * 10^0 grams 1.000000000000000000000000 * 10^0 seconds 4.2 * 10^1 amperes 4.2 * 10^-1 joules 1.23456789012345678901234567890 * 10^5 pascals The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. H - RLE Replacement Problem Statement In JAG Kingdom, ICPC (Intentionally Compressible Programming Code) is one of the common programming languages. Programs in this language only contain uppercase English letters and the same letters often appear repeatedly in ICPC programs. Thus, programmers in JAG Kingdom prefer to compress ICPC programs by Run Length Encoding in order to manage very large-scale ICPC programs. Run Length Encoding (RLE) is a string compression method such that each maximal sequence of the same letters is encoded by a pair of the letter and the length. For example, the string "RRRRLEEE" is represented as "R4L1E3" in RLE. Now, you manage many ICPC programs encoded by RLE. You are developing an editor for ICPC programs encoded by RLE, and now you would like to implement a replacement function. Given three strings $A$, $B$, and $C$ that are encoded by RLE, your task is to implement a function replacing the first occurrence of the substring $B$ in $A$ with $C$, and outputting the edited string encoded by RLE. If $B$ does not occur in $A$, you must output $A$ encoded by RLE without changes. Input The input consists of three lines. > $A$ > $B$ > $C$ The lines represent strings $A$, $B$, and $C$ that are encoded by RLE, respectively. Each of the lines has the following format: > $c_1$ $l_1$ $c_2$ $l_2$ $\ldots$ $c_n$ $l_n$ \$ Each $c_i$ ($1 \leq i \leq n$) is an uppercase English letter (`A`-`Z`) and $l_i$ ($1 \leq i \leq n$, $1 \leq l_i \leq 10^8$) is an integer which represents the length of the repetition of $c_i$. The number $n$ of the pairs of a letter and an integer satisfies $1 \leq n \leq 10^3$. A terminal symbol `$` indicates the end of a string encoded by RLE. The letters and the integers are separated by a single space. It is guaranteed that $c_i \neq c_{i+1}$ holds for any $1 \leq i \leq n-1$. Output Replace the first occurrence of the substring $B$ in $A$ with $C$ if $B$ occurs in $A$, and output the string encoded by RLE. The output must have the following format: > $c_1$ $l_1$ $c_2$ $l_2$ $\ldots$ $c_m$ $l_m$ \$ Here, $c_i \neq c_{i+1}$ for $1 \leq i \leq m-1$ and $l_i \gt 0$ for $1 \leq i \leq m$ must hold. Sample Input 1 R 100 L 20 E 10 \$ R 5 L 10 \$ X 20 \$ Output for the Sample Input 1 R 95 X 20 L 10 E 10 \$ Sample Input 2 A 3 B 3 A 3 \$ A 1 B 3 A 1 \$ A 2 \$ Output for the Sample Input 2 A 6 \$ Example Input R 100 L 20 E 10 \$ R 5 L 10 \$ X 20 \$ Output R 95 X 20 L 10 E 10 \$ The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. A: four tea problem Tea is indispensable for programming contests. Tea has the effect of relieving constant tension [citation needed] There are N players participating in the contest, so I would like to prepare tea for this number of people. There are four types of tea packages, A, B, C, and D, all of which are the same variety but have different contents. For a package X, the price of one package is p_X yen, and it is known that if you buy one, you can make t_X cups of tea. Find the minimum amount needed to make tea for N people. You may have a package that you don't buy at all, and you don't have to buy a package for just N people (if you can make more than N people). Input format The input is given in the following format. N p_A p_B p_C p_D t_A t_B t_C t_D * The first line gives the number of players participating in the contest. * In the second line, the prices of tea in packages A, B, C and D are given separated by blanks. * The third line gives the number of cups of tea that can be made from packages A, B, C, and D, separated by blanks. Constraint * 1 \ leq N \ leq 100 * 1 \ leq p_X \ leq 100 * 1 \ leq t_X \ leq 100 * All inputs are given as integers. Output format Output the minimum amount required to make tea for N people in one line. Input example 1 Ten 1 2 3 4 1 2 4 8 Output example 1 6 * It's best to buy one package B and one D. Input example 2 Five 2 9 9 8 1 4 5 100 Output example 2 8 * You will have 20 times more tea than you need, but buying one Package D is the cheapest way to get more than 5 cups of tea. Input example 3 twenty four 2 3 4 7 7 9 11 20 Output example 3 8 * It's best to buy two packages A and one C. It may not be possible to make just enough tea for the number of people as in this case. Example Input 10 1 2 3 4 1 2 4 8 Output 6 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Gag Segtree has $ N $ of "gags", each with a value of $ V_i $. Segtree decided to publish all the gags in any order. Here, the "joy" you get when you publish the $ i $ th gag to the $ j $ th is expressed as $ V_i --j $. Find the maximum sum of the "joy" you can get. input Input is given from standard input in the following format. $ N $ $ V_1 $ $ V_2 $ $ \ ldots $ $ V_N $ output Please output the maximum value of the sum of "joy". However, the value does not always fit in a 32-bit integer. Insert a line break at the end. Constraint * $ 1 \ leq N \ leq 10 ^ 5 $ * $ 1 \ leq V_i \ leq 10 ^ 5 $ * All inputs are integers. Input example 1 1 59549 Output example 1 59548 Input example 2 Five 2 1 8 5 7 Output example 2 8 Example Input 1 59549 Output 59548 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. For a given array $a_1, a_2, a_3, ... , a_N$ of $N$ elements and $Q$ integers $x_i$ as queries, for each query, print the number of combinations of two integers $(l, r)$ which satisfies the condition: $1 \leq l \leq r \leq N$ and $a_l + a_{l+1} + ... + a_{r-1} + a_r \leq x_i$. Constraints * $1 \leq N \leq 10^5$ * $1 \leq Q \leq 500$ * $1 \leq a_i \leq 10^9$ * $1 \leq x_i \leq 10^{14}$ Input The input is given in the following format. $N$ $Q$ $a_1$ $a_2$ ... $a_N$ $x_1$ $x_2$ ... $x_Q$ Output For each query, print the number of combinations in a line. Example Input 6 5 1 2 3 4 5 6 6 9 12 21 15 Output 9 12 15 21 18 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given a set of n segments on the axis Ox, each segment has integer endpoints between 1 and m inclusive. Segments may intersect, overlap or even coincide with each other. Each segment is characterized by two integers l_i and r_i (1 ≤ l_i ≤ r_i ≤ m) — coordinates of the left and of the right endpoints. Consider all integer points between 1 and m inclusive. Your task is to print all such points that don't belong to any segment. The point x belongs to the segment [l; r] if and only if l ≤ x ≤ r. Input The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100) — the number of segments and the upper bound for coordinates. The next n lines contain two integers each l_i and r_i (1 ≤ l_i ≤ r_i ≤ m) — the endpoints of the i-th segment. Segments may intersect, overlap or even coincide with each other. Note, it is possible that l_i=r_i, i.e. a segment can degenerate to a point. Output In the first line print one integer k — the number of points that don't belong to any segment. In the second line print exactly k integers in any order — the points that don't belong to any segment. All points you print should be distinct. If there are no such points at all, print a single integer 0 in the first line and either leave the second line empty or do not print it at all. Examples Input 3 5 2 2 1 2 5 5 Output 2 3 4 Input 1 7 1 7 Output 0 Note In the first example the point 1 belongs to the second segment, the point 2 belongs to the first and the second segments and the point 5 belongs to the third segment. The points 3 and 4 do not belong to any segment. In the second example all the points from 1 to 7 belong to the first segment. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given n blocks, each of them is of the form [color_1\|value\|color_2], where the block can also be flipped to get [color_2\|value\|color_1]. A sequence of blocks is called valid if the touching endpoints of neighboring blocks have the same color. For example, the sequence of three blocks A, B and C is valid if the left color of the B is the same as the right color of the A and the right color of the B is the same as the left color of C. The value of the sequence is defined as the sum of the values of the blocks in this sequence. Find the maximum possible value of the valid sequence that can be constructed from the subset of the given blocks. The blocks from the subset can be reordered and flipped if necessary. Each block can be used at most once in the sequence. Input The first line of input contains a single integer n (1 ≤ n ≤ 100) — the number of given blocks. Each of the following n lines describes corresponding block and consists of color_{1,i}, value_i and color_{2,i} (1 ≤ color_{1,i}, color_{2,i} ≤ 4, 1 ≤ value_i ≤ 100 000). Output Print exactly one integer — the maximum total value of the subset of blocks, which makes a valid sequence. Examples Input 6 2 1 4 1 2 4 3 4 4 2 8 3 3 16 3 1 32 2 Output 63 Input 7 1 100000 1 1 100000 2 1 100000 2 4 50000 3 3 50000 4 4 50000 4 3 50000 3 Output 300000 Input 4 1 1000 1 2 500 2 3 250 3 4 125 4 Output 1000 Note In the first example, it is possible to form a valid sequence from all blocks. One of the valid sequences is the following: [4\|2\|1] [1\|32\|2] [2\|8\|3] [3\|16\|3] [3\|4\|4] [4\|1\|2] The first block from the input ([2\|1\|4] → [4\|1\|2]) and second ([1\|2\|4] → [4\|2\|1]) are flipped. In the second example, the optimal answers can be formed from the first three blocks as in the following (the second or the third block from the input is flipped): [2\|100000\|1] [1\|100000\|1] [1\|100000\|2] In the third example, it is not possible to form a valid sequence of two or more blocks, so the answer is a sequence consisting only of the first block since it is the block with the largest value. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You have unlimited number of coins with values 1, 2, …, n. You want to select some set of coins having the total value of S. It is allowed to have multiple coins with the same value in the set. What is the minimum number of coins required to get sum S? Input The only line of the input contains two integers n and S (1 ≤ n ≤ 100 000, 1 ≤ S ≤ 10^9) Output Print exactly one integer — the minimum number of coins required to obtain sum S. Examples Input 5 11 Output 3 Input 6 16 Output 3 Note In the first example, some of the possible ways to get sum 11 with 3 coins are: * (3, 4, 4) * (2, 4, 5) * (1, 5, 5) * (3, 3, 5) It is impossible to get sum 11 with less than 3 coins. In the second example, some of the possible ways to get sum 16 with 3 coins are: * (5, 5, 6) * (4, 6, 6) It is impossible to get sum 16 with less than 3 coins. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Petya has a simple graph (that is, a graph without loops or multiple edges) consisting of n vertices and m edges. The weight of the i-th vertex is a_i. The weight of the i-th edge is w_i. A subgraph of a graph is some set of the graph vertices and some set of the graph edges. The set of edges must meet the condition: both ends of each edge from the set must belong to the chosen set of vertices. The weight of a subgraph is the sum of the weights of its edges, minus the sum of the weights of its vertices. You need to find the maximum weight of subgraph of given graph. The given graph does not contain loops and multiple edges. Input The first line contains two numbers n and m (1 ≤ n ≤ 10^3, 0 ≤ m ≤ 10^3) - the number of vertices and edges in the graph, respectively. The next line contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9) - the weights of the vertices of the graph. The following m lines contain edges: the i-e edge is defined by a triple of integers v_i, u_i, w_i (1 ≤ v_i, u_i ≤ n, 1 ≤ w_i ≤ 10^9, v_i ≠ u_i). This triple means that between the vertices v_i and u_i there is an edge of weight w_i. It is guaranteed that the graph does not contain loops and multiple edges. Output Print one integer — the maximum weight of the subgraph of the given graph. Examples Input 4 5 1 5 2 2 1 3 4 1 4 4 3 4 5 3 2 2 4 2 2 Output 8 Input 3 3 9 7 8 1 2 1 2 3 2 1 3 3 Output 0 Note In the first test example, the optimal subgraph consists of the vertices {1, 3, 4} and has weight 4 + 4 + 5 - (1 + 2 + 2) = 8. In the second test case, the optimal subgraph is empty. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given an array a consisting of n integer numbers. You have to color this array in k colors in such a way that: * Each element of the array should be colored in some color; * For each i from 1 to k there should be at least one element colored in the i-th color in the array; * For each i from 1 to k all elements colored in the i-th color should be distinct. Obviously, such coloring might be impossible. In this case, print "NO". Otherwise print "YES" and any coloring (i.e. numbers c_1, c_2, ... c_n, where 1 ≤ c_i ≤ k and c_i is the color of the i-th element of the given array) satisfying the conditions above. If there are multiple answers, you can print any. Input The first line of the input contains two integers n and k (1 ≤ k ≤ n ≤ 5000) — the length of the array a and the number of colors, respectively. The second line of the input contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 5000) — elements of the array a. Output If there is no answer, print "NO". Otherwise print "YES" and any coloring (i.e. numbers c_1, c_2, ... c_n, where 1 ≤ c_i ≤ k and c_i is the color of the i-th element of the given array) satisfying the conditions described in the problem statement. If there are multiple answers, you can print any. Examples Input 4 2 1 2 2 3 Output YES 1 1 2 2 Input 5 2 3 2 1 2 3 Output YES 2 1 1 2 1 Input 5 2 2 1 1 2 1 Output NO Note In the first example the answer 2~ 1~ 2~ 1 is also acceptable. In the second example the answer 1~ 1~ 1~ 2~ 2 is also acceptable. There exist other acceptable answers for both examples. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Sasha and Dima want to buy two n-tier cakes. Each cake should consist of n different tiers: from the size of 1 to the size of n. Tiers should go in order from the smallest to the biggest (from top to bottom). They live on the same street, there are 2 ⋅ n houses in a row from left to right. Each house has a pastry shop where you can buy a cake tier. Unfortunately, in each pastry shop you can buy only one tier of only one specific size: in the i-th house you can buy a tier of the size a_i (1 ≤ a_i ≤ n). Since the guys carry already purchased tiers, and it is impossible to insert a new tier in the middle of the cake, they agreed to buy tiers from the smallest to the biggest. That is, each of them buys tiers in order: 1, then 2, then 3 and so on up to n. Initially, Sasha and Dima are located near the first (leftmost) house. Output the minimum distance that they will have to walk in total to buy both cakes. The distance between any two neighboring houses is exactly 1. Input The first line of the input contains an integer number n — the number of tiers in each cake (1 ≤ n ≤ 10^5). The second line contains 2 ⋅ n integers a_1, a_2, ..., a_{2n} (1 ≤ a_i ≤ n), where a_i is equal to the size of the tier, which can be bought in the i-th house. Remember that in each house you can buy only one tier. It is guaranteed that every number from 1 to n occurs in a exactly two times. Output Print one number — the minimum distance that the guys have to walk in total to buy both cakes. Guys can be near same house at the same time. They begin near the first (leftmost) house. Each of the guys should buy n tiers in ascending order of their sizes. Examples Input 3 1 1 2 2 3 3 Output 9 Input 2 2 1 1 2 Output 5 Input 4 4 1 3 2 2 3 1 4 Output 17 Note In the first example, the possible optimal sequence of actions is: * Sasha buys a tier of size 1 near the 1-st house (a_1=1); * Dima goes to the house 2; * Dima buys a tier of size 1 near the 2-nd house (a_2=1); * Sasha goes to the house 4; * Sasha buys a tier of size 2 near the 4-th house (a_4=2); * Sasha goes to the house 5; * Sasha buys a tier of size 3 near the 5-th house (a_5=3); * Dima goes to the house 3; * Dima buys a tier of size 2 near the 3-rd house (a_3=2); * Dima goes to the house 6; * Dima buys a tier of size 3 near the 6-th house (a_6=3). So, Sasha goes the distance 3+1=4, and Dima goes the distance 1+1+3=5. In total, they cover a distance of 4+5=9. You can make sure that with any other sequence of actions they will walk no less distance. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Bertown has n junctions and m bidirectional roads. We know that one can get from any junction to any other one by the existing roads. As there were more and more cars in the city, traffic jams started to pose real problems. To deal with them the government decided to make the traffic one-directional on all the roads, thus easing down the traffic. Your task is to determine whether there is a way to make the traffic one-directional so that there still is the possibility to get from any junction to any other one. If the answer is positive, you should also find one of the possible ways to orient the roads. Input The first line contains two space-separated integers n and m (2 ≤ n ≤ 105, n - 1 ≤ m ≤ 3·105) which represent the number of junctions and the roads in the town correspondingly. Then follow m lines, each containing two numbers which describe the roads in the city. Each road is determined by two integers ai and bi (1 ≤ ai, bi ≤ n, ai ≠ bi) — the numbers of junctions it connects. It is guaranteed that one can get from any junction to any other one along the existing bidirectional roads. Each road connects different junctions, there is no more than one road between each pair of junctions. Output If there's no solution, print the single number 0. Otherwise, print m lines each containing two integers pi and qi — each road's orientation. That is the traffic flow will move along a one-directional road from junction pi to junction qi. You can print the roads in any order. If there are several solutions to that problem, print any of them. Examples Input 6 8 1 2 2 3 1 3 4 5 4 6 5 6 2 4 3 5 Output 1 2 2 3 3 1 4 5 5 6 6 4 4 2 3 5 Input 6 7 1 2 2 3 1 3 4 5 4 6 5 6 2 4 Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The legendary Farmer John is throwing a huge party, and animals from all over the world are hanging out at his house. His guests are hungry, so he instructs his cow Bessie to bring out the snacks! Moo! There are n snacks flavors, numbered with integers 1, 2, …, n. Bessie has n snacks, one snack of each flavor. Every guest has exactly two favorite flavors. The procedure for eating snacks will go as follows: * First, Bessie will line up the guests in some way. * Then in this order, guests will approach the snacks one by one. * Each guest in their turn will eat all remaining snacks of their favorite flavor. In case no favorite flavors are present when a guest goes up, they become very sad. Help Bessie to minimize the number of sad guests by lining the guests in an optimal way. Input The first line contains integers n and k (2 ≤ n ≤ 10^5, 1 ≤ k ≤ 10^5), the number of snacks and the number of guests. The i-th of the following k lines contains two integers x_i and y_i (1 ≤ x_i, y_i ≤ n, x_i ≠ y_i), favorite snack flavors of the i-th guest. Output Output one integer, the smallest possible number of sad guests. Examples Input 5 4 1 2 4 3 1 4 3 4 Output 1 Input 6 5 2 3 2 1 3 4 6 5 4 5 Output 0 Note In the first example, Bessie can order the guests like this: 3, 1, 2, 4. Guest 3 goes first and eats snacks 1 and 4. Then the guest 1 goes and eats the snack 2 only, because the snack 1 has already been eaten. Similarly, the guest 2 goes up and eats the snack 3 only. All the snacks are gone, so the guest 4 will be sad. In the second example, one optimal ordering is 2, 1, 3, 5, 4. All the guests will be satisfied. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Let's introduce some definitions that will be needed later. Let prime(x) be the set of prime divisors of x. For example, prime(140) = \{ 2, 5, 7 \}, prime(169) = \{ 13 \}. Let g(x, p) be the maximum possible integer p^k where k is an integer such that x is divisible by p^k. For example: * g(45, 3) = 9 (45 is divisible by 3^2=9 but not divisible by 3^3=27), * g(63, 7) = 7 (63 is divisible by 7^1=7 but not divisible by 7^2=49). Let f(x, y) be the product of g(y, p) for all p in prime(x). For example: * f(30, 70) = g(70, 2) ⋅ g(70, 3) ⋅ g(70, 5) = 2^1 ⋅ 3^0 ⋅ 5^1 = 10, * f(525, 63) = g(63, 3) ⋅ g(63, 5) ⋅ g(63, 7) = 3^2 ⋅ 5^0 ⋅ 7^1 = 63. You have integers x and n. Calculate f(x, 1) ⋅ f(x, 2) ⋅ … ⋅ f(x, n) mod{(10^{9} + 7)}. Input The only line contains integers x and n (2 ≤ x ≤ 10^{9}, 1 ≤ n ≤ 10^{18}) — the numbers used in formula. Output Print the answer. Examples Input 10 2 Output 2 Input 20190929 1605 Output 363165664 Input 947 987654321987654321 Output 593574252 Note In the first example, f(10, 1) = g(1, 2) ⋅ g(1, 5) = 1, f(10, 2) = g(2, 2) ⋅ g(2, 5) = 2. In the second example, actual value of formula is approximately 1.597 ⋅ 10^{171}. Make sure you print the answer modulo (10^{9} + 7). In the third example, be careful about overflow issue. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. [Sakuzyo - Imprinting](https://www.youtube.com/watch?v=55Ca6av1kAY) A.R.C. Markland-N is a tall building with n floors numbered from 1 to n. Between each two adjacent floors in the building, there is a staircase connecting them. It's lunchtime for our sensei Colin "ConneR" Neumann Jr, and he's planning for a location to enjoy his meal. ConneR's office is at floor s of the building. On each floor (including floor s, of course), there is a restaurant offering meals. However, due to renovations being in progress, k of the restaurants are currently closed, and as a result, ConneR can't enjoy his lunch there. CooneR wants to reach a restaurant as quickly as possible to save time. What is the minimum number of staircases he needs to walk to reach a closest currently open restaurant. Please answer him quickly, and you might earn his praise and even enjoy the lunch with him in the elegant Neumanns' way! Input The first line contains one integer t (1 ≤ t ≤ 1000) — the number of test cases in the test. Then the descriptions of t test cases follow. The first line of a test case contains three integers n, s and k (2 ≤ n ≤ 10^9, 1 ≤ s ≤ n, 1 ≤ k ≤ min(n-1, 1000)) — respectively the number of floors of A.R.C. Markland-N, the floor where ConneR is in, and the number of closed restaurants. The second line of a test case contains k distinct integers a_1, a_2, …, a_k (1 ≤ a_i ≤ n) — the floor numbers of the currently closed restaurants. It is guaranteed that the sum of k over all test cases does not exceed 1000. Output For each test case print a single integer — the minimum number of staircases required for ConneR to walk from the floor s to a floor with an open restaurant. Example Input 5 5 2 3 1 2 3 4 3 3 4 1 2 10 2 6 1 2 3 4 5 7 2 1 1 2 100 76 8 76 75 36 67 41 74 10 77 Output 2 0 4 0 2 Note In the first example test case, the nearest floor with an open restaurant would be the floor 4. In the second example test case, the floor with ConneR's office still has an open restaurant, so Sensei won't have to go anywhere. In the third example test case, the closest open restaurant is on the 6-th floor. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Tired of boring office work, Denis decided to open a fast food restaurant. On the first day he made a portions of dumplings, b portions of cranberry juice and c pancakes with condensed milk. The peculiarity of Denis's restaurant is the procedure of ordering food. For each visitor Denis himself chooses a set of dishes that this visitor will receive. When doing so, Denis is guided by the following rules: * every visitor should receive at least one dish (dumplings, cranberry juice, pancakes with condensed milk are all considered to be dishes); * each visitor should receive no more than one portion of dumplings, no more than one portion of cranberry juice and no more than one pancake with condensed milk; * all visitors should receive different sets of dishes. What is the maximum number of visitors Denis can feed? Input The first line contains an integer t (1 ≤ t ≤ 500) — the number of test cases to solve. Each of the remaining t lines contains integers a, b and c (0 ≤ a, b, c ≤ 10) — the number of portions of dumplings, the number of portions of cranberry juice and the number of condensed milk pancakes Denis made. Output For each test case print a single integer — the maximum number of visitors Denis can feed. Example Input 7 1 2 1 0 0 0 9 1 7 2 2 3 2 3 2 3 2 2 4 4 4 Output 3 0 4 5 5 5 7 Note In the first test case of the example, Denis can feed the first visitor with dumplings, give the second a portion of cranberry juice, and give the third visitor a portion of cranberry juice and a pancake with a condensed milk. In the second test case of the example, the restaurant Denis is not very promising: he can serve no customers. In the third test case of the example, Denise can serve four visitors. The first guest will receive a full lunch of dumplings, a portion of cranberry juice and a pancake with condensed milk. The second visitor will get only dumplings. The third guest will receive a pancake with condensed milk, and the fourth guest will receive a pancake and a portion of dumplings. Please note that Denis hasn't used all of the prepared products, but is unable to serve more visitors. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The only difference between easy and hard versions is constraints. You are given a sequence a consisting of n positive integers. Let's define a three blocks palindrome as the sequence, consisting of at most two distinct elements (let these elements are a and b, a can be equal b) and is as follows: [\underbrace{a, a, ..., a}_{x}, \underbrace{b, b, ..., b}_{y}, \underbrace{a, a, ..., a}_{x}]. There x, y are integers greater than or equal to 0. For example, sequences [], [2], [1, 1], [1, 2, 1], [1, 2, 2, 1] and [1, 1, 2, 1, 1] are three block palindromes but [1, 2, 3, 2, 1], [1, 2, 1, 2, 1] and [1, 2] are not. Your task is to choose the maximum by length subsequence of a that is a three blocks palindrome. You have to answer t independent test cases. Recall that the sequence t is a a subsequence of the sequence s if t can be derived from s by removing zero or more elements without changing the order of the remaining elements. For example, if s=[1, 2, 1, 3, 1, 2, 1], then possible subsequences are: [1, 1, 1, 1], [3] and [1, 2, 1, 3, 1, 2, 1], but not [3, 2, 3] and [1, 1, 1, 1, 2]. Input The first line of the input contains one integer t (1 ≤ t ≤ 2000) — the number of test cases. Then t test cases follow. The first line of the test case contains one integer n (1 ≤ n ≤ 2000) — the length of a. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 26), where a_i is the i-th element of a. Note that the maximum value of a_i can be up to 26. It is guaranteed that the sum of n over all test cases does not exceed 2000 (∑ n ≤ 2000). Output For each test case, print the answer — the maximum possible length of some subsequence of a that is a three blocks palindrome. Example Input 6 8 1 1 2 2 3 2 1 1 3 1 3 3 4 1 10 10 1 1 26 2 2 1 3 1 1 1 Output 7 2 4 1 1 3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Due to the coronavirus pandemic, city authorities obligated citizens to keep a social distance. The mayor of the city Semyon wants to light up Gluharniki park so that people could see each other even at night to keep the social distance. The park is a rectangular table with n rows and m columns, where the cells of the table are squares, and the boundaries between the cells are streets. External borders are also streets. Every street has length 1. For example, park with n=m=2 has 12 streets. You were assigned to develop a plan for lighting the park. You can put lanterns in the middle of the streets. The lamp lights two squares near it (or only one square if it stands on the border of the park). <image> The park sizes are: n=4, m=5. The lighted squares are marked yellow. Please note that all streets have length 1. Lanterns are placed in the middle of the streets. In the picture not all the squares are lit. Semyon wants to spend the least possible amount of money on lighting but also wants people throughout the park to keep a social distance. So he asks you to find the minimum number of lanterns that are required to light all the squares. Input The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of test cases in the input. Then t test cases follow. Each test case is a line containing two integers n, m (1 ≤ n, m ≤ 10^4) — park sizes. Output Print t answers to the test cases. Each answer must be a single integer — the minimum number of lanterns that are required to light all the squares. Example Input 5 1 1 1 3 2 2 3 3 5 3 Output 1 2 2 5 8 Note Possible optimal arrangement of the lanterns for the 2-nd test case of input data example: <image> Possible optimal arrangement of the lanterns for the 3-rd test case of input data example: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You have n gifts and you want to give all of them to children. Of course, you don't want to offend anyone, so all gifts should be equal between each other. The i-th gift consists of a_i candies and b_i oranges. During one move, you can choose some gift 1 ≤ i ≤ n and do one of the following operations: * eat exactly one candy from this gift (decrease a_i by one); * eat exactly one orange from this gift (decrease b_i by one); * eat exactly one candy and exactly one orange from this gift (decrease both a_i and b_i by one). Of course, you can not eat a candy or orange if it's not present in the gift (so neither a_i nor b_i can become less than zero). As said above, all gifts should be equal. This means that after some sequence of moves the following two conditions should be satisfied: a_1 = a_2 = ... = a_n and b_1 = b_2 = ... = b_n (and a_i equals b_i is not necessary). Your task is to find the minimum number of moves required to equalize all the given gifts. You have to answer t independent test cases. Input The first line of the input contains one integer t (1 ≤ t ≤ 1000) — the number of test cases. Then t test cases follow. The first line of the test case contains one integer n (1 ≤ n ≤ 50) — the number of gifts. The second line of the test case contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the number of candies in the i-th gift. The third line of the test case contains n integers b_1, b_2, ..., b_n (1 ≤ b_i ≤ 10^9), where b_i is the number of oranges in the i-th gift. Output For each test case, print one integer: the minimum number of moves required to equalize all the given gifts. Example Input 5 3 3 5 6 3 2 3 5 1 2 3 4 5 5 4 3 2 1 3 1 1 1 2 2 2 6 1 1000000000 1000000000 1000000000 1000000000 1000000000 1 1 1 1 1 1 3 10 12 8 7 5 4 Output 6 16 0 4999999995 7 Note In the first test case of the example, we can perform the following sequence of moves: * choose the first gift and eat one orange from it, so a = [3, 5, 6] and b = [2, 2, 3]; * choose the second gift and eat one candy from it, so a = [3, 4, 6] and b = [2, 2, 3]; * choose the second gift and eat one candy from it, so a = [3, 3, 6] and b = [2, 2, 3]; * choose the third gift and eat one candy and one orange from it, so a = [3, 3, 5] and b = [2, 2, 2]; * choose the third gift and eat one candy from it, so a = [3, 3, 4] and b = [2, 2, 2]; * choose the third gift and eat one candy from it, so a = [3, 3, 3] and b = [2, 2, 2]. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Some time ago Lesha found an entertaining string s consisting of lowercase English letters. Lesha immediately developed an unique algorithm for this string and shared it with you. The algorithm is as follows. Lesha chooses an arbitrary (possibly zero) number of pairs on positions (i, i + 1) in such a way that the following conditions are satisfied: * for each pair (i, i + 1) the inequality 0 ≤ i < \|s\| - 1 holds; * for each pair (i, i + 1) the equality s_i = s_{i + 1} holds; * there is no index that is contained in more than one pair. After that Lesha removes all characters on indexes contained in these pairs and the algorithm is over. Lesha is interested in the lexicographically smallest strings he can obtain by applying the algorithm to the suffixes of the given string. Input The only line contains the string s (1 ≤ \|s\| ≤ 10^5) — the initial string consisting of lowercase English letters only. Output In \|s\| lines print the lengths of the answers and the answers themselves, starting with the answer for the longest suffix. The output can be large, so, when some answer is longer than 10 characters, instead print the first 5 characters, then "...", then the last 2 characters of the answer. Examples Input abcdd Output 3 abc 2 bc 1 c 0 1 d Input abbcdddeaaffdfouurtytwoo Output 18 abbcd...tw 17 bbcdd...tw 16 bcddd...tw 15 cddde...tw 14 dddea...tw 13 ddeaa...tw 12 deaad...tw 11 eaadf...tw 10 aadfortytw 9 adfortytw 8 dfortytw 9 fdfortytw 8 dfortytw 7 fortytw 6 ortytw 5 rtytw 6 urtytw 5 rtytw 4 tytw 3 ytw 2 tw 1 w 0 1 o Note Consider the first example. * The longest suffix is the whole string "abcdd". Choosing one pair (4, 5), Lesha obtains "abc". * The next longest suffix is "bcdd". Choosing one pair (3, 4), we obtain "bc". * The next longest suffix is "cdd". Choosing one pair (2, 3), we obtain "c". * The next longest suffix is "dd". Choosing one pair (1, 2), we obtain "" (an empty string). * The last suffix is the string "d". No pair can be chosen, so the answer is "d". In the second example, for the longest suffix "abbcdddeaaffdfouurtytwoo" choose three pairs (11, 12), (16, 17), (23, 24) and we obtain "abbcdddeaadfortytw" The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is the hard version of the problem. The difference between the versions is in the number of possible operations that can be made. You can make hacks if and only if you solved both versions of the problem. You are given a binary table of size n × m. This table consists of symbols 0 and 1. You can make such operation: select 3 different cells that belong to one 2 × 2 square and change the symbols in these cells (change 0 to 1 and 1 to 0). Your task is to make all symbols in the table equal to 0. You are allowed to make at most nm operations. You don't need to minimize the number of operations. It can be proved, that it is always possible. Input The first line contains a single integer t (1 ≤ t ≤ 5000) — the number of test cases. The next lines contain descriptions of test cases. The first line of the description of each test case contains two integers n, m (2 ≤ n, m ≤ 100). Each of the next n lines contains a binary string of length m, describing the symbols of the next row of the table. It is guaranteed, that the sum of nm for all test cases does not exceed 20000. Output For each test case print the integer k (0 ≤ k ≤ nm) — the number of operations. In the each of the next k lines print 6 integers x_1, y_1, x_2, y_2, x_3, y_3 (1 ≤ x_1, x_2, x_3 ≤ n, 1 ≤ y_1, y_2, y_3 ≤ m) describing the next operation. This operation will be made with three cells (x_1, y_1), (x_2, y_2), (x_3, y_3). These three cells should be different. These three cells should belong to some 2 × 2 square. Example Input 5 2 2 10 11 3 3 011 101 110 4 4 1111 0110 0110 1111 5 5 01011 11001 00010 11011 10000 2 3 011 101 Output 1 1 1 2 1 2 2 2 2 1 3 1 3 2 1 2 1 3 2 3 4 1 1 1 2 2 2 1 3 1 4 2 3 3 2 4 1 4 2 3 3 4 3 4 4 4 1 2 2 1 2 2 1 4 1 5 2 5 4 1 4 2 5 1 4 4 4 5 3 4 2 1 3 2 2 2 3 1 2 2 1 2 2 Note In the first test case, it is possible to make only one operation with cells (1, 1), (2, 1), (2, 2). After that, all symbols will be equal to 0. In the second test case: * operation with cells (2, 1), (3, 1), (3, 2). After it the table will be: 011 001 000 * operation with cells (1, 2), (1, 3), (2, 3). After it the table will be: 000 000 000 In the fifth test case: * operation with cells (1, 3), (2, 2), (2, 3). After it the table will be: 010 110 * operation with cells (1, 2), (2, 1), (2, 2). After it the table will be: 000 000 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. This is the easy version of the problem. The only difference is that in this version q = 1. You can make hacks only if both versions of the problem are solved. There is a process that takes place on arrays a and b of length n and length n-1 respectively. The process is an infinite sequence of operations. Each operation is as follows: * First, choose a random integer i (1 ≤ i ≤ n-1). * Then, simultaneously set a_i = min\left(a_i, \frac{a_i+a_{i+1}-b_i}{2}\right) and a_{i+1} = max\left(a_{i+1}, \frac{a_i+a_{i+1}+b_i}{2}\right) without any rounding (so values may become non-integer). See notes for an example of an operation. It can be proven that array a converges, i. e. for each i there exists a limit a_i converges to. Let function F(a, b) return the value a_1 converges to after a process on a and b. You are given array b, but not array a. However, you are given a third array c. Array a is good if it contains only integers and satisfies 0 ≤ a_i ≤ c_i for 1 ≤ i ≤ n. Your task is to count the number of good arrays a where F(a, b) ≥ x for q values of x. Since the number of arrays can be very large, print it modulo 10^9+7. Input The first line contains a single integer n (2 ≤ n ≤ 100). The second line contains n integers c_1, c_2 …, c_n (0 ≤ c_i ≤ 100). The third line contains n-1 integers b_1, b_2, …, b_{n-1} (0 ≤ b_i ≤ 100). The fourth line contains a single integer q (q=1). The fifth line contains q space separated integers x_1, x_2, …, x_q (-10^5 ≤ x_i ≤ 10^5). Output Output q integers, where the i-th integer is the answer to the i-th query, i. e. the number of good arrays a where F(a, b) ≥ x_i modulo 10^9+7. Example Input 3 2 3 4 2 1 1 -1 Output 56 Note The following explanation assumes b = [2, 1] and c=[2, 3, 4] (as in the sample). Examples of arrays a that are not good: * a = [3, 2, 3] is not good because a_1 > c_1; * a = [0, -1, 3] is not good because a_2 < 0. One possible good array a is [0, 2, 4]. We can show that no operation has any effect on this array, so F(a, b) = a_1 = 0. Another possible good array a is [0, 1, 4]. In a single operation with i = 1, we set a_1 = min((0+1-2)/(2), 0) and a_2 = max((0+1+2)/(2), 1). So, after a single operation with i = 1, a becomes equal to [-1/2, 3/2, 4]. We can show that no operation has any effect on this array, so F(a, b) = -1/2. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In one one-dimensional world there are n platforms. Platform with index k (platforms are numbered from 1) is a segment with coordinates [(k - 1)m, (k - 1)m + l], and l < m. Grasshopper Bob starts to jump along the platforms from point 0, with each jump he moves exactly d units right. Find out the coordinate of the point, where Bob will fall down. The grasshopper falls down, if he finds himself not on the platform, but if he finds himself on the edge of the platform, he doesn't fall down. Input The first input line contains 4 integer numbers n, d, m, l (1 ≤ n, d, m, l ≤ 106, l < m) — respectively: amount of platforms, length of the grasshopper Bob's jump, and numbers m and l needed to find coordinates of the k-th platform: [(k - 1)m, (k - 1)m + l]. Output Output the coordinates of the point, where the grosshopper will fall down. Don't forget that if Bob finds himself on the platform edge, he doesn't fall down. Examples Input 2 2 5 3 Output 4 Input 5 4 11 8 Output 20 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You've got table a, consisting of n rows, numbered from 1 to n. The i-th line of table a contains ci cells, at that for all i (1 < i ≤ n) holds ci ≤ ci - 1. Let's denote s as the total number of cells of table a, that is, <image>. We know that each cell of the table contains a single integer from 1 to s, at that all written integers are distinct. Let's assume that the cells of the i-th row of table a are numbered from 1 to ci, then let's denote the number written in the j-th cell of the i-th row as ai, j. Your task is to perform several swap operations to rearrange the numbers in the table so as to fulfill the following conditions: 1. for all i, j (1 < i ≤ n; 1 ≤ j ≤ ci) holds ai, j > ai - 1, j; 2. for all i, j (1 ≤ i ≤ n; 1 < j ≤ ci) holds ai, j > ai, j - 1. In one swap operation you are allowed to choose two different cells of the table and swap the recorded there numbers, that is the number that was recorded in the first of the selected cells before the swap, is written in the second cell after it. Similarly, the number that was recorded in the second of the selected cells, is written in the first cell after the swap. Rearrange the numbers in the required manner. Note that you are allowed to perform any number of operations, but not more than s. You do not have to minimize the number of operations. Input The first line contains a single integer n (1 ≤ n ≤ 50) that shows the number of rows in the table. The second line contains n space-separated integers ci (1 ≤ ci ≤ 50; ci ≤ ci - 1) — the numbers of cells on the corresponding rows. Next n lines contain table а. The i-th of them contains ci space-separated integers: the j-th integer in this line represents ai, j. It is guaranteed that all the given numbers ai, j are positive and do not exceed s. It is guaranteed that all ai, j are distinct. Output In the first line print a single integer m (0 ≤ m ≤ s), representing the number of performed swaps. In the next m lines print the description of these swap operations. In the i-th line print four space-separated integers xi, yi, pi, qi (1 ≤ xi, pi ≤ n; 1 ≤ yi ≤ cxi; 1 ≤ qi ≤ cpi). The printed numbers denote swapping the contents of cells axi, yi and api, qi. Note that a swap operation can change the contents of distinct table cells. Print the swaps in the order, in which they should be executed. Examples Input 3 3 2 1 4 3 5 6 1 2 Output 2 1 1 2 2 2 1 3 1 Input 1 4 4 3 2 1 Output 2 1 1 1 4 1 2 1 3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Permutation p is an ordered set of integers p1, p2, ..., pn, consisting of n distinct positive integers, each of them doesn't exceed n. We'll denote the i-th element of permutation p as pi. We'll call number n the size or the length of permutation p1, p2, ..., pn. We'll call position i (1 ≤ i ≤ n) in permutation p1, p2, ..., pn good, if \|p[i] - i\| = 1. Count the number of permutations of size n with exactly k good positions. Print the answer modulo 1000000007 (109 + 7). Input The single line contains two space-separated integers n and k (1 ≤ n ≤ 1000, 0 ≤ k ≤ n). Output Print the number of permutations of length n with exactly k good positions modulo 1000000007 (109 + 7). Examples Input 1 0 Output 1 Input 2 1 Output 0 Input 3 2 Output 4 Input 4 1 Output 6 Input 7 4 Output 328 Note The only permutation of size 1 has 0 good positions. Permutation (1, 2) has 0 good positions, and permutation (2, 1) has 2 positions. Permutations of size 3: 1. (1, 2, 3) — 0 positions 2. <image> — 2 positions 3. <image> — 2 positions 4. <image> — 2 positions 5. <image> — 2 positions 6. (3, 2, 1) — 0 positions The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya’s elder brother Petya loves playing computer games. In one of his favourite computer games Petya reached the final level where a fight with the boss take place. While playing the game Petya found spell scrolls and now he is about to use them. Let’s describe the way fighting goes on this level: 1) The boss has two parameters: max — the initial amount of health and reg — regeneration rate per second. 2) Every scroll also has two parameters: powi — spell power measured in percents — the maximal amount of health counted off the initial one, which allows to use the scroll (i.e. if the boss has more than powi percent of health the scroll cannot be used); and dmgi the damage per second inflicted upon the boss if the scroll is used. As soon as a scroll is used it disappears and another spell is cast upon the boss that inflicts dmgi of damage per second upon him until the end of the game. During the battle the actions per second are performed in the following order: first the boss gets the damage from all the spells cast upon him, then he regenerates reg of health (at the same time he can’t have more than max of health), then the player may use another scroll (no more than one per second). The boss is considered to be defeated if at the end of a second he has nonpositive ( ≤ 0) amount of health. Help Petya to determine whether he can win with the set of scrolls available to him and if he can, determine the minimal number of seconds he needs to do it. Input The first line contains three integers N, max and reg (1 ≤ N, max, reg ≤ 1000) –– the amount of scrolls and the parameters of the boss. The next N lines contain two integers powi and dmgi each — the parameters of the i-th scroll (0 ≤ powi ≤ 100, 1 ≤ dmgi ≤ 2000). Output In case Petya can’t complete this level, output in the single line NO. Otherwise, output on the first line YES. On the second line output the minimal time after which the boss can be defeated and the number of used scrolls. In the next lines for each used scroll output space-separated number of seconds passed from the start of the battle to the moment the scroll was used and the number of the scroll. Scrolls are numbered starting from 1 in the input order. The first scroll is considered to be available to be used after 0 seconds. Output scrolls in the order they were used. It is not allowed to use scrolls after the boss is defeated. Examples Input 2 10 3 100 3 99 1 Output NO Input 2 100 10 100 11 90 9 Output YES 19 2 0 1 10 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There always is something to choose from! And now, instead of "Noughts and Crosses", Inna choose a very unusual upgrade of this game. The rules of the game are given below: There is one person playing the game. Before the beginning of the game he puts 12 cards in a row on the table. Each card contains a character: "X" or "O". Then the player chooses two positive integers a and b (a·b = 12), after that he makes a table of size a × b from the cards he put on the table as follows: the first b cards form the first row of the table, the second b cards form the second row of the table and so on, the last b cards form the last (number a) row of the table. The player wins if some column of the table contain characters "X" on all cards. Otherwise, the player loses. Inna has already put 12 cards on the table in a row. But unfortunately, she doesn't know what numbers a and b to choose. Help her win the game: print to her all the possible ways of numbers a, b that she can choose and win. Input The first line of the input contains integer t (1 ≤ t ≤ 100). This value shows the number of sets of test data in the input. Next follows the description of each of the t tests on a separate line. The description of each test is a string consisting of 12 characters, each character is either "X", or "O". The i-th character of the string shows the character that is written on the i-th card from the start. Output For each test, print the answer to the test on a single line. The first number in the line must represent the number of distinct ways to choose the pair a, b. Next, print on this line the pairs in the format axb. Print the pairs in the order of increasing first parameter (a). Separate the pairs in the line by whitespaces. Examples Input 4 OXXXOXOOXOOX OXOXOXOXOXOX XXXXXXXXXXXX OOOOOOOOOOOO Output 3 1x12 2x6 4x3 4 1x12 2x6 3x4 6x2 6 1x12 2x6 3x4 4x3 6x2 12x1 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The prison of your city has n prisoners. As the prison can't accommodate all of them, the city mayor has decided to transfer c of the prisoners to a prison located in another city. For this reason, he made the n prisoners to stand in a line, with a number written on their chests. The number is the severity of the crime he/she has committed. The greater the number, the more severe his/her crime was. Then, the mayor told you to choose the c prisoners, who will be transferred to the other prison. He also imposed two conditions. They are, * The chosen c prisoners has to form a contiguous segment of prisoners. * Any of the chosen prisoner's crime level should not be greater then t. Because, that will make the prisoner a severe criminal and the mayor doesn't want to take the risk of his running away during the transfer. Find the number of ways you can choose the c prisoners. Input The first line of input will contain three space separated integers n (1 ≤ n ≤ 2·105), t (0 ≤ t ≤ 109) and c (1 ≤ c ≤ n). The next line will contain n space separated integers, the ith integer is the severity ith prisoner's crime. The value of crime severities will be non-negative and will not exceed 109. Output Print a single integer — the number of ways you can choose the c prisoners. Examples Input 4 3 3 2 3 1 1 Output 2 Input 1 1 1 2 Output 0 Input 11 4 2 2 2 0 7 3 2 2 4 9 1 4 Output 6 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One way to create a task is to learn from math. You can generate some random math statement or modify some theorems to get something new and build a new task from that. For example, there is a statement called the "Goldbach's conjecture". It says: "each even number no less than four can be expressed as the sum of two primes". Let's modify it. How about a statement like that: "each integer no less than 12 can be expressed as the sum of two composite numbers." Not like the Goldbach's conjecture, I can prove this theorem. You are given an integer n no less than 12, express it as a sum of two composite numbers. Input The only line contains an integer n (12 ≤ n ≤ 106). Output Output two composite integers x and y (1 < x, y < n) such that x + y = n. If there are multiple solutions, you can output any of them. Examples Input 12 Output 4 8 Input 15 Output 6 9 Input 23 Output 8 15 Input 1000000 Output 500000 500000 Note In the first example, 12 = 4 + 8 and both 4, 8 are composite numbers. You can output "6 6" or "8 4" as well. In the second example, 15 = 6 + 9. Note that you can't output "1 14" because 1 is not a composite number. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Mike has a frog and a flower. His frog is named Xaniar and his flower is named Abol. Initially(at time 0), height of Xaniar is h1 and height of Abol is h2. Each second, Mike waters Abol and Xaniar. <image> So, if height of Xaniar is h1 and height of Abol is h2, after one second height of Xaniar will become <image> and height of Abol will become <image> where x1, y1, x2 and y2 are some integer numbers and <image> denotes the remainder of a modulo b. Mike is a competitive programmer fan. He wants to know the minimum time it takes until height of Xania is a1 and height of Abol is a2. Mike has asked you for your help. Calculate the minimum time or say it will never happen. Input The first line of input contains integer m (2 ≤ m ≤ 106). The second line of input contains integers h1 and a1 (0 ≤ h1, a1 < m). The third line of input contains integers x1 and y1 (0 ≤ x1, y1 < m). The fourth line of input contains integers h2 and a2 (0 ≤ h2, a2 < m). The fifth line of input contains integers x2 and y2 (0 ≤ x2, y2 < m). It is guaranteed that h1 ≠ a1 and h2 ≠ a2. Output Print the minimum number of seconds until Xaniar reaches height a1 and Abol reaches height a2 or print -1 otherwise. Examples Input 5 4 2 1 1 0 1 2 3 Output 3 Input 1023 1 2 1 0 1 2 1 1 Output -1 Note In the first sample, heights sequences are following: Xaniar: <image> Abol: <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. One day Vitaly was going home late at night and wondering: how many people aren't sleeping at that moment? To estimate, Vitaly decided to look which windows are lit in the house he was passing by at that moment. Vitaly sees a building of n floors and 2·m windows on each floor. On each floor there are m flats numbered from 1 to m, and two consecutive windows correspond to each flat. If we number the windows from 1 to 2·m from left to right, then the j-th flat of the i-th floor has windows 2·j - 1 and 2·j in the corresponding row of windows (as usual, floors are enumerated from the bottom). Vitaly thinks that people in the flat aren't sleeping at that moment if at least one of the windows corresponding to this flat has lights on. Given the information about the windows of the given house, your task is to calculate the number of flats where, according to Vitaly, people aren't sleeping. Input The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100) — the number of floors in the house and the number of flats on each floor respectively. Next n lines describe the floors from top to bottom and contain 2·m characters each. If the i-th window of the given floor has lights on, then the i-th character of this line is '1', otherwise it is '0'. Output Print a single integer — the number of flats that have lights on in at least one window, that is, the flats where, according to Vitaly, people aren't sleeping. Examples Input 2 2 0 0 0 1 1 0 1 1 Output 3 Input 1 3 1 1 0 1 0 0 Output 2 Note In the first test case the house has two floors, two flats on each floor. That is, in total there are 4 flats. The light isn't on only on the second floor in the left flat. That is, in both rooms of the flat the light is off. In the second test case the house has one floor and the first floor has three flats. The light is on in the leftmost flat (in both windows) and in the middle flat (in one window). In the right flat the light is off. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given an undirected graph that consists of n vertices and m edges. Initially, each edge is colored either red or blue. Each turn a player picks a single vertex and switches the color of all edges incident to it. That is, all red edges with an endpoint in this vertex change the color to blue, while all blue edges with an endpoint in this vertex change the color to red. Find the minimum possible number of moves required to make the colors of all edges equal. Input The first line of the input contains two integers n and m (1 ≤ n, m ≤ 100 000) — the number of vertices and edges, respectively. The following m lines provide the description of the edges, as the i-th of them contains two integers ui and vi (1 ≤ ui, vi ≤ n, ui ≠ vi) — the indices of the vertices connected by the i-th edge, and a character ci (<image>) providing the initial color of this edge. If ci equals 'R', then this edge is initially colored red. Otherwise, ci is equal to 'B' and this edge is initially colored blue. It's guaranteed that there are no self-loops and multiple edges. Output If there is no way to make the colors of all edges equal output - 1 in the only line of the output. Otherwise first output k — the minimum number of moves required to achieve the goal, then output k integers a1, a2, ..., ak, where ai is equal to the index of the vertex that should be used at the i-th move. If there are multiple optimal sequences of moves, output any of them. Examples Input 3 3 1 2 B 3 1 R 3 2 B Output 1 2 Input 6 5 1 3 R 2 3 R 3 4 B 4 5 R 4 6 R Output 2 3 4 Input 4 5 1 2 R 1 3 R 2 3 B 3 4 B 1 4 B Output -1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Recently, Mike was very busy with studying for exams and contests. Now he is going to chill a bit by doing some sight seeing in the city. City consists of n intersections numbered from 1 to n. Mike starts walking from his house located at the intersection number 1 and goes along some sequence of intersections. Walking from intersection number i to intersection j requires \|i - j\| units of energy. The total energy spent by Mike to visit a sequence of intersections p1 = 1, p2, ..., pk is equal to <image> units of energy. Of course, walking would be boring if there were no shortcuts. A shortcut is a special path that allows Mike walking from one intersection to another requiring only 1 unit of energy. There are exactly n shortcuts in Mike's city, the ith of them allows walking from intersection i to intersection ai (i ≤ ai ≤ ai + 1) (but not in the opposite direction), thus there is exactly one shortcut starting at each intersection. Formally, if Mike chooses a sequence p1 = 1, p2, ..., pk then for each 1 ≤ i < k satisfying pi + 1 = api and api ≠ pi Mike will spend only 1 unit of energy instead of \|pi - pi + 1\| walking from the intersection pi to intersection pi + 1. For example, if Mike chooses a sequence p1 = 1, p2 = ap1, p3 = ap2, ..., pk = apk - 1, he spends exactly k - 1 units of total energy walking around them. Before going on his adventure, Mike asks you to find the minimum amount of energy required to reach each of the intersections from his home. Formally, for each 1 ≤ i ≤ n Mike is interested in finding minimum possible total energy of some sequence p1 = 1, p2, ..., pk = i. Input The first line contains an integer n (1 ≤ n ≤ 200 000) — the number of Mike's city intersection. The second line contains n integers a1, a2, ..., an (i ≤ ai ≤ n , <image>, describing shortcuts of Mike's city, allowing to walk from intersection i to intersection ai using only 1 unit of energy. Please note that the shortcuts don't allow walking in opposite directions (from ai to i). Output In the only line print n integers m1, m2, ..., mn, where mi denotes the least amount of total energy required to walk from intersection 1 to intersection i. Examples Input 3 2 2 3 Output 0 1 2 Input 5 1 2 3 4 5 Output 0 1 2 3 4 Input 7 4 4 4 4 7 7 7 Output 0 1 2 1 2 3 3 Note In the first sample case desired sequences are: 1: 1; m1 = 0; 2: 1, 2; m2 = 1; 3: 1, 3; m3 = \|3 - 1\| = 2. In the second sample case the sequence for any intersection 1 < i is always 1, i and mi = \|1 - i\|. In the third sample case — consider the following intersection sequences: 1: 1; m1 = 0; 2: 1, 2; m2 = \|2 - 1\| = 1; 3: 1, 4, 3; m3 = 1 + \|4 - 3\| = 2; 4: 1, 4; m4 = 1; 5: 1, 4, 5; m5 = 1 + \|4 - 5\| = 2; 6: 1, 4, 6; m6 = 1 + \|4 - 6\| = 3; 7: 1, 4, 5, 7; m7 = 1 + \|4 - 5\| + 1 = 3. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Little Vlad is fond of popular computer game Bota-2. Recently, the developers announced the new add-on named Bota-3. Of course, Vlad immediately bought only to find out his computer is too old for the new game and needs to be updated. There are n video cards in the shop, the power of the i-th video card is equal to integer value ai. As Vlad wants to be sure the new game will work he wants to buy not one, but several video cards and unite their powers using the cutting-edge technology. To use this technology one of the cards is chosen as the leading one and other video cards are attached to it as secondary. For this new technology to work it's required that the power of each of the secondary video cards is divisible by the power of the leading video card. In order to achieve that the power of any secondary video card can be reduced to any integer value less or equal than the current power. However, the power of the leading video card should remain unchanged, i.e. it can't be reduced. Vlad has an infinite amount of money so he can buy any set of video cards. Help him determine which video cards he should buy such that after picking the leading video card and may be reducing some powers of others to make them work together he will get the maximum total value of video power. Input The first line of the input contains a single integer n (1 ≤ n ≤ 200 000) — the number of video cards in the shop. The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 200 000) — powers of video cards. Output The only line of the output should contain one integer value — the maximum possible total power of video cards working together. Examples Input 4 3 2 15 9 Output 27 Input 4 8 2 2 7 Output 18 Note In the first sample, it would be optimal to buy video cards with powers 3, 15 and 9. The video card with power 3 should be chosen as the leading one and all other video cards will be compatible with it. Thus, the total power would be 3 + 15 + 9 = 27. If he buys all the video cards and pick the one with the power 2 as the leading, the powers of all other video cards should be reduced by 1, thus the total power would be 2 + 2 + 14 + 8 = 26, that is less than 27. Please note, that it's not allowed to reduce the power of the leading video card, i.e. one can't get the total power 3 + 1 + 15 + 9 = 28. In the second sample, the optimal answer is to buy all video cards and pick the one with the power 2 as the leading. The video card with the power 7 needs it power to be reduced down to 6. The total power would be 8 + 2 + 2 + 6 = 18. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. After the fourth season Sherlock and Moriary have realized the whole foolishness of the battle between them and decided to continue their competitions in peaceful game of Credit Cards. Rules of this game are simple: each player bring his favourite n-digit credit card. Then both players name the digits written on their cards one by one. If two digits are not equal, then the player, whose digit is smaller gets a flick (knock in the forehead usually made with a forefinger) from the other player. For example, if n = 3, Sherlock's card is 123 and Moriarty's card has number 321, first Sherlock names 1 and Moriarty names 3 so Sherlock gets a flick. Then they both digit 2 so no one gets a flick. Finally, Sherlock names 3, while Moriarty names 1 and gets a flick. Of course, Sherlock will play honestly naming digits one by one in the order they are given, while Moriary, as a true villain, plans to cheat. He is going to name his digits in some other order (however, he is not going to change the overall number of occurences of each digit). For example, in case above Moriarty could name 1, 2, 3 and get no flicks at all, or he can name 2, 3 and 1 to give Sherlock two flicks. Your goal is to find out the minimum possible number of flicks Moriarty will get (no one likes flicks) and the maximum possible number of flicks Sherlock can get from Moriarty. Note, that these two goals are different and the optimal result may be obtained by using different strategies. Input The first line of the input contains a single integer n (1 ≤ n ≤ 1000) — the number of digits in the cards Sherlock and Moriarty are going to use. The second line contains n digits — Sherlock's credit card number. The third line contains n digits — Moriarty's credit card number. Output First print the minimum possible number of flicks Moriarty will get. Then print the maximum possible number of flicks that Sherlock can get from Moriarty. Examples Input 3 123 321 Output 0 2 Input 2 88 00 Output 2 0 Note First sample is elaborated in the problem statement. In the second sample, there is no way Moriarty can avoid getting two flicks. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Erelong Leha was bored by calculating of the greatest common divisor of two factorials. Therefore he decided to solve some crosswords. It's well known that it is a very interesting occupation though it can be very difficult from time to time. In the course of solving one of the crosswords, Leha had to solve a simple task. You are able to do it too, aren't you? Leha has two strings s and t. The hacker wants to change the string s at such way, that it can be found in t as a substring. All the changes should be the following: Leha chooses one position in the string s and replaces the symbol in this position with the question mark "?". The hacker is sure that the question mark in comparison can play the role of an arbitrary symbol. For example, if he gets string s="ab?b" as a result, it will appear in t="aabrbb" as a substring. Guaranteed that the length of the string s doesn't exceed the length of the string t. Help the hacker to replace in s as few symbols as possible so that the result of the replacements can be found in t as a substring. The symbol "?" should be considered equal to any other symbol. Input The first line contains two integers n and m (1 ≤ n ≤ m ≤ 1000) — the length of the string s and the length of the string t correspondingly. The second line contains n lowercase English letters — string s. The third line contains m lowercase English letters — string t. Output In the first line print single integer k — the minimal number of symbols that need to be replaced. In the second line print k distinct integers denoting the positions of symbols in the string s which need to be replaced. Print the positions in any order. If there are several solutions print any of them. The numbering of the positions begins from one. Examples Input 3 5 abc xaybz Output 2 2 3 Input 4 10 abcd ebceabazcd Output 1 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Even if the world is full of counterfeits, I still regard it as wonderful. Pile up herbs and incense, and arise again from the flames and ashes of its predecessor — as is known to many, the phoenix does it like this. The phoenix has a rather long lifespan, and reincarnates itself once every a! years. Here a! denotes the factorial of integer a, that is, a! = 1 × 2 × ... × a. Specifically, 0! = 1. Koyomi doesn't care much about this, but before he gets into another mess with oddities, he is interested in the number of times the phoenix will reincarnate in a timespan of b! years, that is, <image>. Note that when b ≥ a this value is always integer. As the answer can be quite large, it would be enough for Koyomi just to know the last digit of the answer in decimal representation. And you're here to provide Koyomi with this knowledge. Input The first and only line of input contains two space-separated integers a and b (0 ≤ a ≤ b ≤ 1018). Output Output one line containing a single decimal digit — the last digit of the value that interests Koyomi. Examples Input 2 4 Output 2 Input 0 10 Output 0 Input 107 109 Output 2 Note In the first example, the last digit of <image> is 2; In the second example, the last digit of <image> is 0; In the third example, the last digit of <image> is 2. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Students Vasya and Petya are studying at the BSU (Byteland State University). At one of the breaks they decided to order a pizza. In this problem pizza is a circle of some radius. The pizza was delivered already cut into n pieces. The i-th piece is a sector of angle equal to ai. Vasya and Petya want to divide all pieces of pizza into two continuous sectors in such way that the difference between angles of these sectors is minimal. Sector angle is sum of angles of all pieces in it. Pay attention, that one of sectors can be empty. Input The first line contains one integer n (1 ≤ n ≤ 360) — the number of pieces into which the delivered pizza was cut. The second line contains n integers ai (1 ≤ ai ≤ 360) — the angles of the sectors into which the pizza was cut. The sum of all ai is 360. Output Print one integer — the minimal difference between angles of sectors that will go to Vasya and Petya. Examples Input 4 90 90 90 90 Output 0 Input 3 100 100 160 Output 40 Input 1 360 Output 360 Input 4 170 30 150 10 Output 0 Note In first sample Vasya can take 1 and 2 pieces, Petya can take 3 and 4 pieces. Then the answer is \|(90 + 90) - (90 + 90)\| = 0. In third sample there is only one piece of pizza that can be taken by only one from Vasya and Petya. So the answer is \|360 - 0\| = 360. In fourth sample Vasya can take 1 and 4 pieces, then Petya will take 2 and 3 pieces. So the answer is \|(170 + 10) - (30 + 150)\| = 0. Picture explaning fourth sample: <image> Both red and green sectors consist of two adjacent pieces of pizza. So Vasya can take green sector, then Petya will take red sector. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Jamie is preparing a Codeforces round. He has got an idea for a problem, but does not know how to solve it. Help him write a solution to the following problem: Find k integers such that the sum of two to the power of each number equals to the number n and the largest integer in the answer is as small as possible. As there may be multiple answers, you are asked to output the lexicographically largest one. To be more clear, consider all integer sequence with length k (a1, a2, ..., ak) with <image>. Give a value <image> to each sequence. Among all sequence(s) that have the minimum y value, output the one that is the lexicographically largest. For definitions of powers and lexicographical order see notes. Input The first line consists of two integers n and k (1 ≤ n ≤ 1018, 1 ≤ k ≤ 105) — the required sum and the length of the sequence. Output Output "No" (without quotes) in a single line if there does not exist such sequence. Otherwise, output "Yes" (without quotes) in the first line, and k numbers separated by space in the second line — the required sequence. It is guaranteed that the integers in the answer sequence fit the range [ - 1018, 1018]. Examples Input 23 5 Output Yes 3 3 2 1 0 Input 13 2 Output No Input 1 2 Output Yes -1 -1 Note Sample 1: 23 + 23 + 22 + 21 + 20 = 8 + 8 + 4 + 2 + 1 = 23 Answers like (3, 3, 2, 0, 1) or (0, 1, 2, 3, 3) are not lexicographically largest. Answers like (4, 1, 1, 1, 0) do not have the minimum y value. Sample 2: It can be shown there does not exist a sequence with length 2. Sample 3: <image> Powers of 2: If x > 0, then 2x = 2·2·2·...·2 (x times). If x = 0, then 2x = 1. If x < 0, then <image>. Lexicographical order: Given two different sequences of the same length, (a1, a2, ... , ak) and (b1, b2, ... , bk), the first one is smaller than the second one for the lexicographical order, if and only if ai < bi, for the first i where ai and bi differ. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. In distant future on Earth day lasts for n hours and that's why there are n timezones. Local times in adjacent timezones differ by one hour. For describing local time, hours numbers from 1 to n are used, i.e. there is no time "0 hours", instead of it "n hours" is used. When local time in the 1-st timezone is 1 hour, local time in the i-th timezone is i hours. Some online programming contests platform wants to conduct a contest that lasts for an hour in such a way that its beginning coincides with beginning of some hour (in all time zones). The platform knows, that there are ai people from i-th timezone who want to participate in the contest. Each person will participate if and only if the contest starts no earlier than s hours 00 minutes local time and ends not later than f hours 00 minutes local time. Values s and f are equal for all time zones. If the contest starts at f hours 00 minutes local time, the person won't participate in it. Help platform select such an hour, that the number of people who will participate in the contest is maximum. Input The first line contains a single integer n (2 ≤ n ≤ 100 000) — the number of hours in day. The second line contains n space-separated integers a1, a2, ..., an (1 ≤ ai ≤ 10 000), where ai is the number of people in the i-th timezone who want to participate in the contest. The third line contains two space-separated integers s and f (1 ≤ s < f ≤ n). Output Output a single integer — the time of the beginning of the contest (in the first timezone local time), such that the number of participants will be maximum possible. If there are many answers, output the smallest among them. Examples Input 3 1 2 3 1 3 Output 3 Input 5 1 2 3 4 1 1 3 Output 4 Note In the first example, it's optimal to start competition at 3 hours (in first timezone). In this case, it will be 1 hour in the second timezone and 2 hours in the third timezone. Only one person from the first timezone won't participate. In second example only people from the third and the fourth timezones will participate. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya is a regular participant at programming contests and is already experienced in finding important sentences in long statements. Of course, numbers constraints are important — factorization of a number less than 1000000 is easier than of a number less than 1000000000. However, sometimes it's hard to understand the number at the first glance. Could it be shortened? For example, instead of 1000000 you could write 10^{6}, instead of 1000000000 —10^{9}, instead of 1000000007 — 10^{9}+7. Vasya decided that, to be concise, the notation should follow several rules: * the notation should only consist of numbers, operations of addition ("+"), multiplication ("") and exponentiation ("^"), in particular, the use of braces is forbidden; the use of several exponentiation operations in a row is forbidden, for example, writing "2^3^4" is unacceptable; * the value of the resulting expression equals to the initial number; * the notation should consist of the minimal amount of symbols. Given n, find the equivalent concise notation for it. Input The only line contains a single integer n (1 ≤ n ≤ 10 000 000 000). Output Output a concise notation of the number n. If there are several concise notations, output any of them. Examples Input 2018 Output 2018 Input 1000000007 Output 10^9+7 Input 10000000000 Output 100^5 Input 2000000000 Output 2*10^9 Note The third sample allows the answer 10^10 also of the length 5. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Little Arihant has always wanted to be the best Pokemon trainer in this world. And he thinks he has achieved his goal, so he wants to quickly go and meet Professor Oak and verify this fact. But like all Pokemon trainers, he has a weird habit, too. He catches Pokemons which can go through evolution to become a better one. After roaming in the Pokeworld for days, he has finally managed to catch k such Pokemons. The way he can make a Pokemon go through evolution is NOT by making them fight battles, but by using an evolution stone. Since he has k Pokemons, he naturally needs k evolution stones for every one of them, as well. Now it takes little Arihant one complete day, to use the evolution stone on one Pokemon. And for each Pokemon, he knows how many days will they take to evolute after the evolution stone has been used on them. He will go to meet Professor Oak, the very next day, once all his Pokemons have gone through evolution. He can select the order of applying evolution stones as he likes, so he wants to do it in such a way that he gets to meet Professor Oak as soon as possible! Input Format: The input has two lines. The first line will contain an integer k, which denotes the number of Pokemons. Then, a line with k integers follows, where the i-th integer denotes the number of days it takes for the i-th Pokemon to evolve. Output Format: You need to print the earliest day when little Arihant can go meet Professor Oak. Constraints: The days are numbered 1, 2, 3, 4, 5, 6... 1 ≤ k ≤ 10^5. 1 ≤ Number of days ≤ 10^6. SAMPLE INPUT 2 3 1 SAMPLE OUTPUT 5 Explanation Here's how it happens: Day 1: He uses the evolution stone on the Pokemon which takes 3 days. Day 2: 3 days Pokemon's first day. He uses the evolution stone on the Pokemon which takes 1 day. Day 3: 3 days Pokemon's second day. 1 day Pokemon's first and final day. It goes through evolution. Day 4: 3 days Pokemon's third day. It goes through evolution. All Pokemons done with, he can go to Professor Oak, now. Day 5: He goes to Professor Oak. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. After minting loads of money from innocent people by using the psychic powers (As we saw in the previous question!) - little Jhool managed to woo his girlfriend big Jhool! And now, he's in love - madly, madly in love with his girlfriend. But the road to love is not easy as they say, little Jhool and big Jhool live in two different cities, and they often have to rely on multiple online clients to communicate. And many times, due to frustration, little Jhool starts hitting his keyboard, pressing the wrong keys - and thus, he ends up sending her a meaningless message, which frustrates his girlfriend - obviously. But big Jhool is a smart girl and she knows what to reciprocate back to her boyfriend's message based on the message she has received from him. She figures that if and only if, she can delete several characters from the message sent by her boyfriend, resulting in the word "love", it will be considered that she managed to understand his message, and if not... well, some trouble is waiting for her boyfriend! Input format: There's only one line in the input, it contains the message sent by her boyfriend. Output format: If she managed to understand his message, print "I love you, too!" - else, print "Let us breakup!" Constraints: 1 ≤ Length of message ≤ 100 Example: Let's say that the message sent by him is: ahhellolvloe, the answer would be "I love you, too!" since it is possible that by deleting characters, she gets the word love. Let's say that the message sent by him is: lvoe, the answer would be, "Let us breakup!" because there's no way you could get the word love by only using the delete operation. PS: You can only perform deletion, on the message sent - you cannot rearrange the words in the message! SAMPLE INPUT lov3333333asdafajfgnkdfn33333e SAMPLE OUTPUT I love you, too! Explanation You can delete characters to get the word love. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. See Russian Translation It's a fine sunny afternoon today in California. Looking at the pleasant weather, Sumit is all ready to go out and play with his friend Rohil. Unfortunately, Rohil is down with fever. Seeing that his friend is ill, Sumit decides not to go out - instead play with Rohil inside the house. Sumit loves math, on the contrary Rohil loves strings. Sumit decides to play a game that involves more of strings and less of Maths so that Rohil could be at ease and have fun playing it. The game is simple and is played on a piece of paper. Sumit writes down a long list of names on that paper and passes it to Rohil. Rohil gets confused on seeing so many names on that paper and asks Sumit about the game. So, Sumit explains him the rules of the game. Rohil is supposed to partition the names into groups, such that: Each name belongs to exactly one group. Names that belong to the same group are pairwise anagrams. The first character of all the names in the same group are equal. The last character of all the names in the same group are equal. The number of groups is minimum possible. Note: Two strings are called anagrams if it's possible to form one string from the other by changing the order of its characters. Rohil would have won the game easily, if he would have been fit and fine but since he is ill right now he needs your help in winning the game. So, help out Rohil and do give him your blessings. Input: The first line contains a single integer N indicating the size of the list. This is followed by N lines where each line contains a name listed by Sumit. Output: In a single line print minimum number of groups in a partition that satisfy above conditions Constraints: 1 ≤ N ≤100 1 ≤ Length of a name ≤ 100 All names will consist of lowercase English alphabets(a-z). SAMPLE INPUT 6 vinay vainy vinit viint avinash aasivnh SAMPLE OUTPUT 3 Explanation There are 3 special groups 1) vinay and vainy 2) vinit and viint 3) avinash and aavsinh The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Given an integer a as input, print the value a + a^2 + a^3. Constraints * 1 \leq a \leq 10 * a is an integer. Input Input is given from Standard Input in the following format: a Output Print the value a + a^2 + a^3 as an integer. Examples Input 2 Output 14 Input 10 Output 1110 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have a tree G with N vertices numbered 1 to N. The i-th edge of G connects Vertex a_i and Vertex b_i. Consider adding zero or more edges in G, and let H be the graph resulted. Find the number of graphs H that satisfy the following conditions, modulo 998244353. * H does not contain self-loops or multiple edges. * The diameters of G and H are equal. * For every pair of vertices in H that is not directly connected by an edge, the addition of an edge directly connecting them would reduce the diameter of the graph. Constraints * 3 \le N \le 2 \times 10^5 * 1 \le a_i, b_i \le N * The given graph is a tree. Input Input is given from Standard Input in the following format: N a_1 b_1 \vdots a_{N-1} b_{N-1} Output Print the answer. Examples Input 6 1 6 2 1 5 2 3 4 2 3 Output 3 Input 3 1 2 2 3 Output 1 Input 9 1 2 2 3 4 2 1 7 6 1 2 5 5 9 6 8 Output 27 Input 19 2 4 15 8 1 16 1 3 12 19 1 18 7 11 11 15 12 9 1 6 7 14 18 2 13 12 13 5 16 13 7 1 11 10 7 17 Output 78732 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are N slimes lining up from left to right. The colors of these slimes will be given as a string S of length N consisting of lowercase English letters. The i-th slime from the left has the color that corresponds to the i-th character of S. Adjacent slimes with the same color will fuse into one larger slime without changing the color. If there were a slime adjacent to this group of slimes before fusion, that slime is now adjacent to the new larger slime. Ultimately, how many slimes will be there? Constraints * 1 \leq N \leq 10^5 * \|S\| = N * S consists of lowercase English letters. Input Input is given from Standard Input in the following format: N S Output Print the final number of slimes. Examples Input 10 aabbbbaaca Output 5 Input 5 aaaaa Output 1 Input 20 xxzaffeeeeddfkkkkllq Output 10 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Snuke found a random number generator. It generates an integer between 0 and 2^N-1 (inclusive). An integer sequence A_0, A_1, \cdots, A_{2^N-1} represents the probability that each of these integers is generated. The integer i (0 \leq i \leq 2^N-1) is generated with probability A_i / S, where S = \sum_{i=0}^{2^N-1} A_i. The process of generating an integer is done independently each time the generator is executed. Snuke has an integer X, which is now 0. He can perform the following operation any number of times: * Generate an integer v with the generator and replace X with X \oplus v, where \oplus denotes the bitwise XOR. For each integer i (0 \leq i \leq 2^N-1), find the expected number of operations until X becomes i, and print it modulo 998244353. More formally, represent the expected number of operations as an irreducible fraction P/Q. Then, there exists a unique integer R such that R \times Q \equiv P \mod 998244353,\ 0 \leq R < 998244353, so print this R. We can prove that, for every i, the expected number of operations until X becomes i is a finite rational number, and its integer representation modulo 998244353 can be defined. Constraints * 1 \leq N \leq 18 * 1 \leq A_i \leq 1000 * All values in input are integers. Input Input is given from Standard Input in the following format: N A_0 A_1 \cdots A_{2^N-1} Output Print 2^N lines. The (i+1)-th line (0 \leq i \leq 2^N-1) should contain the expected number of operations until X becomes i, modulo 998244353. Examples Input 2 1 1 1 1 Output 0 4 4 4 Input 2 1 2 1 2 Output 0 499122180 4 499122180 Input 4 337 780 799 10 796 875 331 223 941 67 148 483 390 565 116 355 Output 0 468683018 635850749 96019779 657074071 24757563 745107950 665159588 551278361 143136064 557841197 185790407 988018173 247117461 129098626 789682908 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Kenkoooo is planning a trip in Republic of Snuke. In this country, there are n cities and m trains running. The cities are numbered 1 through n, and the i-th train connects City u_i and v_i bidirectionally. Any city can be reached from any city by changing trains. Two currencies are used in the country: yen and snuuk. Any train fare can be paid by both yen and snuuk. The fare of the i-th train is a_i yen if paid in yen, and b_i snuuk if paid in snuuk. In a city with a money exchange office, you can change 1 yen into 1 snuuk. However, when you do a money exchange, you have to change all your yen into snuuk. That is, if Kenkoooo does a money exchange when he has X yen, he will then have X snuuk. Currently, there is a money exchange office in every city, but the office in City i will shut down in i years and can never be used in and after that year. Kenkoooo is planning to depart City s with 10^{15} yen in his pocket and head for City t, and change his yen into snuuk in some city while traveling. It is acceptable to do the exchange in City s or City t. Kenkoooo would like to have as much snuuk as possible when he reaches City t by making the optimal choices for the route to travel and the city to do the exchange. For each i=0,...,n-1, find the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i years. You can assume that the trip finishes within the year. Constraints * 2 \leq n \leq 10^5 * 1 \leq m \leq 10^5 * 1 \leq s,t \leq n * s \neq t * 1 \leq u_i < v_i \leq n * 1 \leq a_i,b_i \leq 10^9 * If i\neq j, then u_i \neq u_j or v_i \neq v_j. * Any city can be reached from any city by changing trains. * All values in input are integers. Input Input is given from Standard Input in the following format: n m s t u_1 v_1 a_1 b_1 : u_m v_m a_m b_m Output Print n lines. In the i-th line, print the maximum amount of snuuk that Kenkoooo has when he reaches City t if he goes on a trip from City s to City t after i-1 years. Examples Input 4 3 2 3 1 4 1 100 1 2 1 10 1 3 20 1 Output 999999999999998 999999999999989 999999999999979 999999999999897 Input 8 12 3 8 2 8 685087149 857180777 6 7 298270585 209942236 2 4 346080035 234079976 2 5 131857300 22507157 4 8 30723332 173476334 2 6 480845267 448565596 1 4 181424400 548830121 4 5 57429995 195056405 7 8 160277628 479932440 1 6 475692952 203530153 3 5 336869679 160714712 2 7 389775999 199123879 Output 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 999999574976994 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. We have N sticks with negligible thickness. The length of the i-th stick is A_i. Snuke wants to select four different sticks from these sticks and form a rectangle (including a square), using the sticks as its sides. Find the maximum possible area of the rectangle. Constraints * 4 \leq N \leq 10^5 * 1 \leq A_i \leq 10^9 * A_i is an integer. Input Input is given from Standard Input in the following format: N A_1 A_2 ... A_N Output Print the maximum possible area of the rectangle. If no rectangle can be formed, print 0. Examples Input 6 3 1 2 4 2 1 Output 2 Input 4 1 2 3 4 Output 0 Input 10 3 3 3 3 4 4 4 5 5 5 Output 20 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Every day, N passengers arrive at Takahashi Airport. The i-th passenger arrives at time T_i. Every passenger arrived at Takahashi airport travels to the city by bus. Each bus can accommodate up to C passengers. Naturally, a passenger cannot take a bus that departs earlier than the airplane arrives at the airport. Also, a passenger will get angry if he/she is still unable to take a bus K units of time after the arrival of the airplane. For that reason, it is necessary to arrange buses so that the i-th passenger can take a bus departing at time between T_i and T_i + K (inclusive). When setting the departure times for buses under this condition, find the minimum required number of buses. Here, the departure time for each bus does not need to be an integer, and there may be multiple buses that depart at the same time. Constraints * 2 \leq N \leq 100000 * 1 \leq C \leq 10^9 * 1 \leq K \leq 10^9 * 1 \leq T_i \leq 10^9 * C, K and T_i are integers. Input The input is given from Standard Input in the following format: N C K T_1 T_2 : T_N Output Print the minimum required number of buses. Examples Input 5 3 5 1 2 3 6 12 Output 3 Input 6 3 3 7 6 2 8 10 6 Output 3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are N rabbits on a number line. The rabbits are conveniently numbered 1 through N. The coordinate of the initial position of rabbit i is x_i. The rabbits will now take exercise on the number line, by performing sets described below. A set consists of M jumps. The j-th jump of a set is performed by rabbit a_j (2≤a_j≤N-1). For this jump, either rabbit a_j-1 or rabbit a_j+1 is chosen with equal probability (let the chosen rabbit be rabbit x), then rabbit a_j will jump to the symmetric point of its current position with respect to rabbit x. The rabbits will perform K sets in succession. For each rabbit, find the expected value of the coordinate of its eventual position after K sets are performed. Constraints * 3≤N≤10^5 * x_i is an integer. * \|x_i\|≤10^9 * 1≤M≤10^5 * 2≤a_j≤N-1 * 1≤K≤10^{18} Input The input is given from Standard Input in the following format: N x_1 x_2 ... x_N M K a_1 a_2 ... a_M Output Print N lines. The i-th line should contain the expected value of the coordinate of the eventual position of rabbit i after K sets are performed. The output is considered correct if the absolute or relative error is at most 10^{-9}. Examples Input 3 -1 0 2 1 1 2 Output -1.0 1.0 2.0 Input 3 1 -1 1 2 2 2 2 Output 1.0 -1.0 1.0 Input 5 0 1 3 6 10 3 10 2 3 4 Output 0.0 3.0 7.0 8.0 10.0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There is a puzzle to complete by combining 14 numbers from 1 to 9. Complete by adding another number to the given 13 numbers. The conditions for completing the puzzle are * You must have one combination of the same numbers. * The remaining 12 numbers are 4 combinations of 3 numbers. The combination of three numbers is either three of the same numbers or three consecutive numbers. However, sequences such as 9 1 2 are not considered consecutive numbers. * The same number can be used up to 4 times. Create a program that reads a string of 13 numbers and outputs all the numbers that can complete the puzzle in ascending order. If you cannot complete the puzzle by adding any number from 1 to 9, output 0. For example, if the given string is 3456666777999 If there is a "2", 234 567 666 77 999 If there is a "3", then 33 456 666 777 999 If there is a "5", then 345 567 666 77 999 If there is an "8", then 345 666 678 77 999 And so on, the puzzle is complete when one of the numbers 2 3 5 8 is added. Note that "6" is fine, but it will be used for the 5th time, so it cannot be used in this example. Input The input consists of multiple datasets. For each dataset, 13 numbers are given on one line. The number of datasets does not exceed 50. Output For each dataset, the numbers that can complete the puzzle are output on one line in ascending order, separated by blanks. Example Input 3649596966777 6358665788577 9118992346175 9643871425498 7755542764533 1133557799246 Output 2 3 5 8 3 4 1 2 3 4 5 6 7 8 9 7 8 9 1 2 3 4 6 7 8 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Decimal numbers are a common notation system currently in use and use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent all numbers. Binary numbers are a popular notation in the computer world and use two symbols, 0 and 1, to represent all numbers. Only the four numbers 0, 1, 2, and 3 are used in quaternary numbers. In quaternary numbers, when the number is incremented from 0, it will be carried to the next digit when it reaches 4. Therefore, the decimal number 4 is carried to the expression "10". Decimal \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| ... --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \|- --- Binary \| 0 \| 1 \| 10 \| 11 \| 100 \| 101 \| 110 \| 111 \| 1000 \| 101 \| 1010 \| ... Quadrant \| 0 \| 1 \| 2 \| 3 \| 10 \| 11 \| 12 \| 13 \| 20 \| 21 \| 22 \| ... In Hawaii, fish and taro were counted between fingers in the old days, so it seems that they used quaternary numbers instead of decimal numbers. Create a program that converts the integer n input in decimal to decimal and outputs it. Input A sequence of multiple datasets is given as input. The end of the input is indicated by a single line of -1. One integer n (0 ≤ n ≤ 1000000) is given on one row for each dataset. The number of datasets does not exceed 2000. Output The result of conversion to quaternary number for each input data set is output on one line. Example Input 7 4 0 12 10 10000 -1 Output 13 10 0 30 22 2130100 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The appearance of the sun is called "sunrise" and the hiding is called "sunset". What is the exact time when the sun is on the horizon? As shown in the figure below, we will represent the sun as a circle and the horizon as a straight line. At this time, the time of "sunrise" and "sunset" of the sun is the moment when the upper end of the circle representing the sun coincides with the straight line representing the horizon. After sunrise, the daytime is when the top of the circle is above the straight line, and nighttime is when the circle is completely hidden below the straight line. <image> Create a program that inputs the height from the horizon to the center of the sun at a certain time and the radius of the sun, and outputs whether the time is "daytime", "sunrise or sunset", or "nighttime". Input The input is given in the following format. H R The input consists of one line and is given the integer H (-1000 ≤ H ≤ 1000), which represents the height from the horizon at a certain time to the center of the sun, and the integer R (1 ≤ R ≤ 1000), which represents the radius. However, H is 0 when the center of the sun is on the horizon, positive when it is above it, and negative when it is below it. Output Outputs "1" in the daytime, "0" in the sunrise or sunset, and "-1" in the nighttime on one line. Examples Input -3 3 Output 0 Input 3 3 Output 1 Input -4 3 Output -1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. As the first step in algebra, students learn quadratic formulas and their factorization. Often, the factorization is a severe burden for them. A large number of students cannot master the factorization; such students cannot be aware of the elegance of advanced algebra. It might be the case that the factorization increases the number of people who hate mathematics. Your job here is to write a program which helps students of an algebra course. Given a quadratic formula, your program should report how the formula can be factorized into two linear formulas. All coefficients of quadratic formulas and those of resultant linear formulas are integers in this problem. The coefficients a, b and c of a quadratic formula ax2 + bx + c are given. The values of a, b and c are integers, and their absolute values do not exceed 10000. From these values, your program is requested to find four integers p, q, r and s, such that ax2 + bx + c = (px + q)(rx + s). Since we are considering integer coefficients only, it is not always possible to factorize a quadratic formula into linear formulas. If the factorization of the given formula is impossible, your program should report that fact. Input The input is a sequence of lines, each representing a quadratic formula. An input line is given in the following format. > a b c Each of a, b and c is an integer. They satisfy the following inequalities. > 0 < a <= 10000 > -10000 <= b <= 10000 > -10000 <= c <= 10000 The greatest common divisor of a, b and c is 1. That is, there is no integer k, greater than 1, such that all of a, b and c are divisible by k. The end of input is indicated by a line consisting of three 0's. Output For each input line, your program should output the four integers p, q, r and s in a line, if they exist. These integers should be output in this order, separated by one or more white spaces. If the factorization is impossible, your program should output a line which contains the string "Impossible" only. The following relations should hold between the values of the four coefficients. > p > 0 > r > 0 > (p > r) or (p = r and q >= s) These relations, together with the fact that the greatest common divisor of a, b and c is 1, assure the uniqueness of the solution. If you find a way to factorize the formula, it is not necessary to seek another way to factorize it. Example Input 2 5 2 1 1 1 10 -7 0 1 0 -3 0 0 0 Output 2 1 1 2 Impossible 10 -7 1 0 Impossible The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Professor Pathfinder is a distinguished authority on the structure of hyperlinks in the World Wide Web. For establishing his hypotheses, he has been developing software agents, which automatically traverse hyperlinks and analyze the structure of the Web. Today, he has gotten an intriguing idea to improve his software agents. However, he is very busy and requires help from good programmers. You are now being asked to be involved in his development team and to create a small but critical software module of his new type of software agents. Upon traversal of hyperlinks, Pathfinder’s software agents incrementally generate a map of visited portions of the Web. So the agents should maintain the list of traversed hyperlinks and visited web pages. One problem in keeping track of such information is that two or more different URLs can point to the same web page. For instance, by typing any one of the following five URLs, your favorite browsers probably bring you to the same web page, which as you may have visited is the home page of the ACM ICPC Ehime contest. http://www.ehime-u.ac.jp/ICPC/ http://www.ehime-u.ac.jp/ICPC http://www.ehime-u.ac.jp/ICPC/../ICPC/ http://www.ehime-u.ac.jp/ICPC/./ http://www.ehime-u.ac.jp/ICPC/index.html Your program should reveal such aliases for Pathfinder’s experiments. Well, . . . but it were a real challenge and to be perfect you might have to embed rather compli- cated logic into your program. We are afraid that even excellent programmers like you could not complete it in five hours. So, we make the problem a little simpler and subtly unrealis- tic. You should focus on the path parts (i.e. /ICPC/, /ICPC, /ICPC/../ICPC/, /ICPC/./, and /ICPC/index.html in the above example) of URLs and ignore the scheme parts (e.g. http://), the server parts (e.g. www.ehime-u.ac.jp), and other optional parts. You should carefully read the rules described in the sequel since some of them may not be based on the reality of today’s Web and URLs. Each path part in this problem is an absolute pathname, which specifies a path from the root directory to some web page in a hierarchical (tree-shaped) directory structure. A pathname always starts with a slash (/), representing the root directory, followed by path segments delim- ited by a slash. For instance, /ICPC/index.html is a pathname with two path segments ICPC and index.html. All those path segments but the last should be directory names and the last one the name of an ordinary file where a web page is stored. However, we have one exceptional rule: an ordinary file name index.html at the end of a pathname may be omitted. For instance, a pathname /ICPC/index.html can be shortened to /ICPC/, if index.html is an existing ordinary file name. More precisely, if ICPC is the name of an existing directory just under the root and index.html is the name of an existing ordinary file just under the /ICPC directory, /ICPC/index.html and /ICPC/ refer to the same web page. Furthermore, the last slash following the last path segment can also be omitted. That is, for instance, /ICPC/ can be further shortened to /ICPC. However, /index.html can only be abbreviated to / (a single slash). You should pay special attention to path segments consisting of a single period (.) or a double period (..), both of which are always regarded as directory names. The former represents the directory itself and the latter represents its parent directory. Therefore, if /ICPC/ refers to some web page, both /ICPC/./ and /ICPC/../ICPC/ refer to the same page. Also /ICPC2/../ICPC/ refers to the same page if ICPC2 is the name of an existing directory just under the root; otherwise it does not refer to any web page. Note that the root directory does not have any parent directory and thus such pathnames as /../ and /ICPC/../../index.html cannot point to any web page. Your job in this problem is to write a program that checks whether two given pathnames refer to existing web pages and, if so, examines whether they are the same. Input The input consists of multiple datasets. The first line of each dataset contains two positive integers N and M, both of which are less than or equal to 100 and are separated by a single space character. The rest of the dataset consists of N + 2M lines, each of which contains a syntactically correct pathname of at most 100 characters. You may assume that each path segment enclosed by two slashes is of length at least one. In other words, two consecutive slashes cannot occur in any pathname. Each path segment does not include anything other than alphanumerical characters (i.e. ‘a’-‘z’, ‘A’-‘Z’, and ‘0’-‘9’) and periods (‘.’). The first N pathnames enumerate all the web pages (ordinary files). Every existing directory name occurs at least once in these pathnames. You can assume that these pathnames do not include any path segments consisting solely of single or double periods and that the last path segments are ordinary file names. Therefore, you do not have to worry about special rules for index.html and single/double periods. You can also assume that no two of the N pathnames point to the same page. Each of the following M pairs of pathnames is a question: do the two pathnames point to the same web page? These pathnames may include single or double periods and may be terminated by a slash. They may include names that do not correspond to existing directories or ordinary files. Two zeros in a line indicate the end of the input. Output For each dataset, your program should output the M answers to the M questions, each in a separate line. Each answer should be “yes” if both point to the same web page, “not found” if at least one of the pathnames does not point to any one of the first N web pages listed in the input, or “no” otherwise. Example Input 5 6 /home/ACM/index.html /ICPC/index.html /ICPC/general.html /ICPC/japanese/index.html /ICPC/secret/confidential/2005/index.html /home/ACM/ /home/ICPC/../ACM/ /ICPC/secret/ /ICPC/secret/index.html /ICPC /ICPC/../ICPC/index.html /ICPC /ICPC/general.html /ICPC/japanese/.././ /ICPC/japanese/./../ /home/ACM/index.html /home/ACM/index.html/ 1 4 /index.html/index.html / /index.html/index.html /index.html /index.html/index.html /.. /index.html/../.. /index.html/ /index.html/index.html/.. 0 0 Output not found not found yes no yes not found not found yes not found not found The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Look for the Winner! The citizens of TKB City are famous for their deep love in elections and vote counting. Today they hold an election for the next chairperson of the electoral commission. Now the voting has just been closed and the counting is going to start. The TKB citizens have strong desire to know the winner as early as possible during vote counting. The election candidate receiving the most votes shall be the next chairperson. Suppose for instance that we have three candidates A, B, and C and ten votes. Suppose also that we have already counted six of the ten votes and the vote counts of A, B, and C are four, one, and one, respectively. At this moment, every candidate has a chance to receive four more votes and so everyone can still be the winner. However, if the next vote counted is cast for A, A is ensured to be the winner since A already has five votes and B or C can have at most four votes at the end. In this example, therefore, the TKB citizens can know the winner just when the seventh vote is counted. Your mission is to write a program that receives every vote counted, one by one, identifies the winner, and determines when the winner gets ensured. Input The input consists of at most 1500 datasets, each consisting of two lines in the following format. n c1 c2 … cn n in the first line represents the number of votes, and is a positive integer no greater than 100. The second line represents the n votes, separated by a space. Each ci (1 ≤ i ≤ n) is a single uppercase letter, i.e. one of 'A' through 'Z'. This represents the election candidate for which the i-th vote was cast. Counting shall be done in the given order from c1 to cn. You should assume that at least two stand as candidates even when all the votes are cast for one candidate. The end of the input is indicated by a line containing a zero. Output For each dataset, unless the election ends in a tie, output a single line containing an uppercase letter c and an integer d separated by a space: c should represent the election winner and d should represent after counting how many votes the winner is identified. Otherwise, that is, if the election ends in a tie, output a single line containing `TIE'. Sample Input 1 A 4 A A B B 5 L M N L N 6 K K K K K K 6 X X X Y Z X 10 A A A B A C A C C B 10 U U U U U V V W W W 0 Output for the Sample Input A 1 TIE TIE K 4 X 5 A 7 U 8 Example Input 1 A 4 A A B B 5 L M N L N 6 K K K K K K 6 X X X Y Z X 10 A A A B A C A C C B 10 U U U U U V V W W W 0 Output A 1 TIE TIE K 4 X 5 A 7 U 8 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. International Car Production Company (ICPC), one of the largest automobile manufacturers in the world, is now developing a new vehicle called "Two-Wheel Buggy". As its name suggests, the vehicle has only two wheels. Quite simply, "Two-Wheel Buggy" is made up of two wheels (the left wheel and the right wheel) and a axle (a bar connecting two wheels). The figure below shows its basic structure. <image> Figure 7: The basic structure of the buggy Before making a prototype of this new vehicle, the company decided to run a computer simula- tion. The details of the simulation is as follows. In the simulation, the buggy will move on the x-y plane. Let D be the distance from the center of the axle to the wheels. At the beginning of the simulation, the center of the axle is at (0, 0), the left wheel is at (-D, 0), and the right wheel is at (D, 0). The radii of two wheels are 1. <image> Figure 8: The initial position of the buggy The movement of the buggy in the simulation is controlled by a sequence of instructions. Each instruction consists of three numbers, Lspeed, Rspeed and time. Lspeed and Rspeed indicate the rotation speed of the left and right wheels, respectively, expressed in degree par second. time indicates how many seconds these two wheels keep their rotation speed. If a speed of a wheel is positive, it will rotate in the direction that causes the buggy to move forward. Conversely, if a speed is negative, it will rotate in the opposite direction. For example, if we set Lspeed as -360, the left wheel will rotate 360-degree in one second in the direction that makes the buggy move backward. We can set Lspeed and Rspeed differently, and this makes the buggy turn left or right. Note that we can also set one of them positive and the other negative (in this case, the buggy will spin around). <image> Figure 9: Examples Your job is to write a program that calculates the final position of the buggy given a instruction sequence. For simplicity, you can can assume that wheels have no width, and that they would never slip. Input The input consists of several datasets. Each dataset is formatted as follows. N D Lspeed1 Rspeed1 time1 . . . Lspeedi Rspeedi timei . . . LspeedN RspeedN timeN The first line of a dataset contains two positive integers, N and D (1 ≤ N ≤ 100, 1 ≤ D ≤ 10). N indicates the number of instructions in the dataset, and D indicates the distance between the center of axle and the wheels. The following N lines describe the instruction sequence. The i-th line contains three integers, Lspeedi, i, and timei (-360 ≤ Lspeedi, Rspeedi ≤ 360, 1 ≤ timei ), describing the i-th instruction to the buggy. You can assume that the sum of timei is at most 500. The end of input is indicated by a line containing two zeros. This line is not part of any dataset and hence should not be processed. Output For each dataset, output two lines indicating the final position of the center of the axle. The first line should contain the x-coordinate, and the second line should contain the y-coordinate. The absolute error should be less than or equal to 10-3 . No extra character should appear in the output. Example Input 1 1 180 90 2 1 1 180 180 20 2 10 360 -360 5 -90 360 8 3 2 100 60 9 -72 -72 10 -45 -225 5 0 0 Output 3.00000 3.00000 0.00000 62.83185 12.00000 0.00000 -2.44505 13.12132 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. The University of Aizu Elementary School (Aizu University and Small) is famous as one of Japan's leading competition programmer training schools. Of course, it is essential to practice the algorithm even when attending an athletic meet. Of course you, the director of the competitive programming department, want to win this tournament as well. This time we will focus on a certain competition. A certain competition is a traditional competition held in Aizu, large and small. There are V cones in the schoolyard. Several pairs of cones are connected by arrows drawn with white lines. The tip of the arrow is attached to only one side, and an integer is also written. The same pair of cones may be connected by multiple arrows. The player arbitrarily selects a cone and starts moving. The movement moves from the cone where the competitor is located to the next cone by moving in that direction on the arrow. You may follow the same cone and the same arrow many times. After moving from cone to cone, the competitor can choose to move further or end the move. The purpose of this competition is to make the score K or higher by following the arrows. As for the score, the integer value written together is added each time the arrow is followed. A player with a score of K or higher with fewer arrows passing through wins. If the number of arrows is the same, the player with the higher score wins. Output how many arrows should be taken when the athlete makes the optimum movement according to this rule. Also, if the number of arrows that have passed through is 100 or less, output all the cones that should be passed through in order. There may be multiple optimum movements, but the result of any movement may be output. If there is no movement to make the score K or higher, output -1. In addition, all cones are numbered from 0 to V-1, and all colors are green (meaningful). Constraints The input satisfies the following conditions. * All inputs are integers * 2 ≤ V ≤ 150 * 0 ≤ E ≤ V × V * 0 <K ≤ 106 * 0 ≤ vi1, vi2 <V (0 <i ≤ E) * vi1 ≠ vi2 (0 <i ≤ E) * 0 <ci ≤ 100 (0 <i ≤ E) * Some inputs contain i, j such that vi1 = vj1 and vi2 = vj2 (i ≠ j, 0 <i, j ≤ E). Input The input is given in the following format. > V E K > v11 v12 c1 > ... > vi1 vi2 ci > ... > vE1 vE2 cE > here, * V is the number of cones * E is the number of arrows * vi, 1 is the cone number of the starting point of the arrow i * vi2 is the cone number at the end of the arrow i * ci is an integer indicated by the arrow Is. Output The output consists of two lines. * 1st line Output with the number of arrows passing through when the optimum movement is performed * 2nd line Outputs in the order of passing through the cone number to be passed, separated by blanks There may be multiple optimum movements, but the result of any movement may be output. If the number of arrows to be passed exceeds 100, do not output the second line. If there is no optimal movement to make the score K or higher, -1 should be output on the first line and nothing should be output on the second line. Examples Input 3 9 89 2 0 2 1 0 3 2 0 1 2 0 3 0 1 1 0 1 2 1 2 3 0 1 1 1 0 2 Output 34 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 0 Input 2 0 1 Output -1 Input 7 8 4000 0 1 1 1 0 1 1 2 2 2 3 2 3 4 2 5 4 3 3 5 4 5 6 5 Output 3991 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Problem Statement In the headquarter building of ICPC (International Company of Plugs & Connectors), there are $M$ light bulbs and they are controlled by $N$ switches. Each light bulb can be turned on or off by exactly one switch. Each switch may control multiple light bulbs. When you operate a switch, all the light bulbs controlled by the switch change their states. You lost the table that recorded the correspondence between the switches and the light bulbs, and want to restore it. You decided to restore the correspondence by the following procedure. * At first, every switch is off and every light bulb is off. * You operate some switches represented by $S_1$. * You check the states of the light bulbs represented by $B_1$. * You operate some switches represented by $S_2$. * You check the states of the light bulbs represented by $B_2$. * ... * You operate some switches represented by $S_Q$. * You check the states of the light bulbs represented by $B_Q$. After you operate some switches and check the states of the light bulbs, the states of the switches and the light bulbs are kept for next operations. Can you restore the correspondence between the switches and the light bulbs using the information about the switches you have operated and the states of the light bulbs you have checked? Input The input consists of multiple datasets. The number of dataset is no more than $50$ and the file size is no more than $10\mathrm{MB}$. Each dataset is formatted as follows. > $N$ $M$ $Q$ > $S_1$ $B_1$ > : > : > $S_Q$ $B_Q$ The first line of each dataset contains three integers $N$ ($1 \le N \le 36$), $M$ ($1 \le M \le 1{,}000$), $Q$ ($0 \le Q \le 1{,}000$), which denote the number of switches, the number of light bulbs and the number of operations respectively. The following $Q$ lines describe the information about the switches you have operated and the states of the light bulbs you have checked. The $i$-th of them contains two strings $S_i$ and $B_i$ of lengths $N$ and $M$ respectively. Each $S_i$ denotes the set of the switches you have operated: $S_{ij}$ is either $0$ or $1$, which denotes the $j$-th switch is not operated or operated respectively. Each $B_i$ denotes the states of the light bulbs: $B_{ij}$ is either $0$ or $1$, which denotes the $j$-th light bulb is off or on respectively. You can assume that there exists a correspondence between the switches and the light bulbs which is consistent with the given information. The end of input is indicated by a line containing three zeros. Output For each dataset, output the correspondence between the switches and the light bulbs consisting of $M$ numbers written in base-$36$. In the base-$36$ system for this problem, the values $0$-$9$ and $10$-$35$ are represented by the characters '0'-'9' and 'A'-'Z' respectively. The $i$-th character of the correspondence means the number of the switch controlling the $i$-th light bulb. If you cannot determine which switch controls the $i$-th light bulb, output '?' as the $i$-th character instead of the number of a switch. Sample Input 3 10 3 000 0000000000 110 0000001111 101 1111111100 2 2 0 1 1 0 2 1 1 01 1 11 11 10 10000000000 10000000000 11000000000 01000000000 01100000000 00100000000 00110000000 00010000000 00011000000 00001000000 00001100000 00000100000 00000110000 00000010000 00000011000 00000001000 00000001100 00000000100 00000000110 00000000010 0 0 0 Output for the Sample Input 2222221100 ?? 0 1 0123456789A Example Input 3 10 3 000 0000000000 110 0000001111 101 1111111100 2 2 0 1 1 0 2 1 1 01 1 11 11 10 10000000000 10000000000 11000000000 01000000000 01100000000 00100000000 00110000000 00010000000 00011000000 00001000000 00001100000 00000100000 00000110000 00000010000 00000011000 00000001000 00000001100 00000000100 00000000110 00000000010 0 0 0 Output 2222221100 ?? 0 1 0123456789A The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. jfen There is a one-person game to play on the H × W board. This game is a game to move 0 or 1 balls in each cell. You were playing this game and found it difficult to move the ball accurately from cell to cell because the ball is so round. So you decided to make a robot that would move the ball as instructed. Here, the notation (y, x) is used to represent the cell. This represents the cell in the xth column of the yth row. Instructions to the robot are given by the four integers a, b, c, d. This means moving the ball at (a, b) to (c, d). At this time, it is guaranteed that the ball exists in (a, b) and does not exist in (c, d). The state of the board is expressed by the following notation called "jfen". [Data in the first line] / [Data in the second line] /.../ [Data in the H line] This notation is a slash-separated concatenation of data representing each line on the board. The data in each row is represented by a character string, and the state of the cells in the 1st to Wth columns is described in order from the left. This string is represented by a number and the letter'b', where the integer represented by the number represents the number of consecutive blank cells and'b' represents the cell in which the ball resides. Here, integers must not be consecutive in the data of each row. Also, the sum of the integers written in the data of each line and the sum of the number of'b'are always W. As an example, consider the following board surface condition. A blank cell is represented by a'.', And a cell in which the ball exists is represented by a'b'. The j character from the left of the i-th row in this example represents the state of the cell in the j-th column of the i-th row on the board. .... .b.b .... The above board is expressed in jfen as follows. 4 / 1b1b / 4 Create a program that outputs the board surface after the robot moves the ball when the robot is given the current board surface state and a command to move one ball. At this time, output the board surface in jfen notation. Input The input consists of multiple datasets. Each dataset has the following format. > S > a b c d Each dataset consists of two lines, and the first line is given the jfen-formatted string S, which represents the state of the board. Here, the size of the board satisfies 2 ≤ W and H ≤ 9. Instructions to the robot are given to the following lines. This represents an instruction to move the ball at (a, b) to (c, d). The end of the input is represented by #. Output The output is one line of character string that expresses the state of the board after executing the instruction for each data set in jfen notation. There must be no other characters on the output line. Sample Input b1 / 1b 1 1 1 2 b5 / bbbbbb 2 4 1 4 b2b2b / 7 1 4 2 4 Output for Sample Input 1b / 1b b2b2 / bbb1bb b5b / 3b3 The initial state of the first input is as follows. b. .b Since the ball of (1,1) is moved to (1,2), the board surface after the command is as follows. .b .b Example Input b1/1b 1 1 1 2 b5/bbbbbb 2 4 1 4 b2b2b/7 1 4 2 4 # Output 1b/1b b2b2/bbb1bb b5b/3b3 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Generalized leap year Normally, whether or not the year x is a leap year is defined as follows. 1. If x is a multiple of 400, it is a leap year. 2. Otherwise, if x is a multiple of 100, it is not a leap year. 3. Otherwise, if x is a multiple of 4, it is a leap year. 4. If not, it is not a leap year. This can be generalized as follows. For a sequence A1, ..., An, we define whether the year x is a "generalized leap year" as follows. 1. For the smallest i (1 ≤ i ≤ n) such that x is a multiple of Ai, if i is odd, it is a generalized leap year, and if it is even, it is not a generalized leap year. 2. When such i does not exist, it is not a generalized leap year if n is odd, but a generalized leap year if n is even. For example, when A = [400, 100, 4], the generalized leap year for A is equivalent to a normal leap year. Given the sequence A1, ..., An and the positive integers l, r. Answer the number of positive integers x such that l ≤ x ≤ r such that year x is a generalized leap year for A. Input The input consists of up to 50 datasets. Each dataset is represented in the following format. > n l r A1 A2 ... An The integer n satisfies 1 ≤ n ≤ 50. The integers l and r satisfy 1 ≤ l ≤ r ≤ 4000. For each i, the integer Ai satisfies 1 ≤ Ai ≤ 4000. The end of the input is represented by a line of three zeros. Output Print the answer in one line for each dataset. Sample Input 3 1988 2014 400 100 Four 1 1000 1999 1 2 1111 3333 2 2 6 2000 3000 Five 7 11 9 3 13 0 0 0 Output for the Sample Input 7 1000 2223 785 Example Input 3 1988 2014 400 100 4 1 1000 1999 1 2 1111 3333 2 2 6 2000 3000 5 7 11 9 3 13 0 0 0 Output 7 1000 2223 785 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Problem There are $ N $ streetlights on a two-dimensional square of $ W \ times H $. Gaccho wants to start with $ (1,1) $ and go to $ (W, H) $. Gaccho is afraid of dark places, so he only wants to walk in the squares that are brightened by the streetlights. Initially, all streetlights only brighten the squares with the streetlights. So, Gaccho decided to set the cost $ r_i $ for his favorite streetlight $ i $. There may be street lights for which no cost is set. By consuming the cost $ r_i $, the streetlight $ i $ can brighten the range within $ r_i $ in Manhattan distance around the streetlight. However, the cost is a positive integer. Gaccho can move to the adjacent square in either the up, down, left, or right direction. Gaccho decided to set the total value of $ r_i $ to be the minimum. Find the total value at that time. The Manhattan distance between two points $ (a, b) $ and $ (c, d) $ is represented by $ \| a−c \| $ + $ \| b−d \| $. Constraints The input satisfies the following conditions. * $ 1 \ leq W \ leq 500 $ * $ 1 \ leq H \ leq 500 $ * $ 1 \ leq N \ leq 100 $ * $ 1 \ leq N \ leq W \ times H $ * $ 1 \ leq $$ x_i $$ \ leq W $ * $ 1 \ leq $$ y_i $$ \ leq H $ * There are no multiple streetlights at the same coordinates Input The input is given in the following format. $ W $ $ H $ $ N $ $ x_1 $ $ y_1 $ ... $ x_N $ $ y_N $ All inputs are given as integers. $ W $, $ H $, and $ N $ are given on the first line, separated by blanks. In the following $ N $ line, the coordinates $ ($$ x_i $, $ y_i $$) $ of the streetlight $ i $ are given, separated by blanks. Output Output the minimum value of the total value of $ r_i $ on one line. Examples Input 10 10 1 6 6 Output 10 Input 5 10 3 3 9 2 8 5 1 Output 8 Input 1 1 1 1 1 Output 0 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Write a program which reads an integer and prints sum of its digits. Input The input consists of multiple datasets. For each dataset, an integer x is given in a line. The number of digits in x does not exceed 1000. The input ends with a line including single zero. Your program should not process for this terminal symbol. Output For each dataset, print the sum of digits in x. Example Input 123 55 1000 0 Output 6 10 1 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Lira is now very keen on compiler development. :) She knows that one of the most important components of a compiler, is its parser. A parser is, in simple terms, a software component that processes text, and checks it's semantic correctness, or, if you prefer, if the text is properly built. As an example, in declaring and initializing an integer, in C/C++, you can't do something like: int = x ;4 as the semantics of such statement is incorrect, as we all know that the datatype must precede an identifier and only afterwards should come the equal sign and the initialization value, so, the corrected statement should be: int x = 4; Today, Lira is concerned with an abstract instruction which is composed of the characters "<" and ">" , which she will use on the design of her language, L++ :D. She is using it as an abstraction for generating XML code Tags in an easier fashion and she understood that, for an expression to be valid, a "<" symbol must always have a corresponding ">" character somewhere (not necessary immediately) after it. Moreover, each ">" symbol should correspond to exactly one "<" symbol. So, for instance, the instructions: <<>> <> <><> are all valid. While: >> ><>< are not. Given some expressions which represent some instructions to be analyzed by Lira's compiler, you should tell the length of the longest prefix of each of these expressions that is valid, or 0 if there's no such a prefix. Input Input will consist of an integer T denoting the number of test cases to follow. Then, T strings follow, each on a single line, representing a possible expression in L++. Output For each expression you should output the length of the longest prefix that is valid or 0 if there's no such a prefix. Constraints 1 ≤ T ≤ 500 1 ≤ The length of a single expression ≤ 10^6 The total size all the input expressions is no more than 5*10^6 Example Input: 3 <<>> >< <>>> Output: 4 0 2 The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There is a beautiful garden of stones in Innopolis. Its most beautiful place is the n piles with stones numbered from 1 to n. EJOI participants have visited this place twice. When they first visited it, the number of stones in piles was x_1, x_2, …, x_n, correspondingly. One of the participants wrote down this sequence in a notebook. They visited it again the following day, and the number of stones in piles was equal to y_1, y_2, …, y_n. One of the participants also wrote it down in a notebook. It is well known that every member of the EJOI jury during the night either sits in the room 108 or comes to the place with stones. Each jury member who comes there either takes one stone for himself or moves one stone from one pile to another. We can assume that there is an unlimited number of jury members. No one except the jury goes to the place with stones at night. Participants want to know whether their notes can be correct or they are sure to have made a mistake. Input The first line of the input file contains a single integer n, the number of piles with stones in the garden (1 ≤ n ≤ 50). The second line contains n integers separated by spaces x_1, x_2, …, x_n, the number of stones in piles recorded in the notebook when the participants came to the place with stones for the first time (0 ≤ x_i ≤ 1000). The third line contains n integers separated by spaces y_1, y_2, …, y_n, the number of stones in piles recorded in the notebook when the participants came to the place with stones for the second time (0 ≤ y_i ≤ 1000). Output If the records can be consistent output "Yes", otherwise output "No" (quotes for clarity). Examples Input 5 1 2 3 4 5 2 1 4 3 5 Output Yes Input 5 1 1 1 1 1 1 0 1 0 1 Output Yes Input 3 2 3 9 1 7 9 Output No Note In the first example, the following could have happened during the night: one of the jury members moved one stone from the second pile to the first pile, and the other jury member moved one stone from the fourth pile to the third pile. In the second example, the jury took stones from the second and fourth piles. It can be proved that it is impossible for the jury members to move and took stones to convert the first array into the second array. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. You are given two arrays a and b of positive integers, with length n and m respectively. Let c be an n × m matrix, where c_{i,j} = a_i ⋅ b_j. You need to find a subrectangle of the matrix c such that the sum of its elements is at most x, and its area (the total number of elements) is the largest possible. Formally, you need to find the largest number s such that it is possible to choose integers x_1, x_2, y_1, y_2 subject to 1 ≤ x_1 ≤ x_2 ≤ n, 1 ≤ y_1 ≤ y_2 ≤ m, (x_2 - x_1 + 1) × (y_2 - y_1 + 1) = s, and $$$∑_{i=x_1}^{x_2}{∑_{j=y_1}^{y_2}{c_{i,j}}} ≤ x.$$$ Input The first line contains two integers n and m (1 ≤ n, m ≤ 2000). The second line contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 2000). The third line contains m integers b_1, b_2, …, b_m (1 ≤ b_i ≤ 2000). The fourth line contains a single integer x (1 ≤ x ≤ 2 ⋅ 10^{9}). Output If it is possible to choose four integers x_1, x_2, y_1, y_2 such that 1 ≤ x_1 ≤ x_2 ≤ n, 1 ≤ y_1 ≤ y_2 ≤ m, and ∑_{i=x_1}^{x_2}{∑_{j=y_1}^{y_2}{c_{i,j}}} ≤ x, output the largest value of (x_2 - x_1 + 1) × (y_2 - y_1 + 1) among all such quadruplets, otherwise output 0. Examples Input 3 3 1 2 3 1 2 3 9 Output 4 Input 5 1 5 4 2 4 5 2 5 Output 1 Note Matrix from the first sample and the chosen subrectangle (of blue color): <image> Matrix from the second sample and the chosen subrectangle (of blue color): <image> The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Vasya is reading a e-book. The file of the book consists of n pages, numbered from 1 to n. The screen is currently displaying the contents of page x, and Vasya wants to read the page y. There are two buttons on the book which allow Vasya to scroll d pages forwards or backwards (but he cannot scroll outside the book). For example, if the book consists of 10 pages, and d = 3, then from the first page Vasya can scroll to the first or to the fourth page by pressing one of the buttons; from the second page — to the first or to the fifth; from the sixth page — to the third or to the ninth; from the eighth — to the fifth or to the tenth. Help Vasya to calculate the minimum number of times he needs to press a button to move to page y. Input The first line contains one integer t (1 ≤ t ≤ 10^3) — the number of testcases. Each testcase is denoted by a line containing four integers n, x, y, d (1≤ n, d ≤ 10^9, 1 ≤ x, y ≤ n) — the number of pages, the starting page, the desired page, and the number of pages scrolled by pressing one button, respectively. Output Print one line for each test. If Vasya can move from page x to page y, print the minimum number of times he needs to press a button to do it. Otherwise print -1. Example Input 3 10 4 5 2 5 1 3 4 20 4 19 3 Output 4 -1 5 Note In the first test case the optimal sequence is: 4 → 2 → 1 → 3 → 5. In the second test case it is possible to get to pages 1 and 5. In the third test case the optimal sequence is: 4 → 7 → 10 → 13 → 16 → 19. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. There are n segments [l_i, r_i] for 1 ≤ i ≤ n. You should divide all segments into two non-empty groups in such way that there is no pair of segments from different groups which have at least one common point, or say that it's impossible to do it. Each segment should belong to exactly one group. To optimize testing process you will be given multitest. Input The first line contains one integer T (1 ≤ T ≤ 50000) — the number of queries. Each query contains description of the set of segments. Queries are independent. First line of each query contains single integer n (2 ≤ n ≤ 10^5) — number of segments. It is guaranteed that ∑{n} over all queries does not exceed 10^5. The next n lines contains two integers l_i, r_i per line (1 ≤ l_i ≤ r_i ≤ 2 ⋅ 10^5) — the i-th segment. Output For each query print n integers t_1, t_2, ..., t_n (t_i ∈ \{1, 2\}) — for each segment (in the same order as in the input) t_i equals 1 if the i-th segment will belongs to the first group and 2 otherwise. If there are multiple answers, you can print any of them. If there is no answer, print -1. Example Input 3 2 5 5 2 3 3 3 5 2 3 2 3 3 3 3 4 4 5 5 Output 2 1 -1 1 1 2 Note In the first query the first and the second segments should be in different groups, but exact numbers don't matter. In the second query the third segment intersects with the first and the second segments, so they should be in the same group, but then the other group becomes empty, so answer is -1. In the third query we can distribute segments in any way that makes groups non-empty, so any answer of 6 possible is correct. The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.
Solve the programming task below in a Python markdown code block. Little Petya loves presents. His mum bought him two strings of the same size for his birthday. The strings consist of uppercase and lowercase Latin letters. Now Petya wants to compare those two strings lexicographically. The letters' case does not matter, that is an uppercase letter is considered equivalent to the corresponding lowercase letter. Help Petya perform the comparison. Input Each of the first two lines contains a bought string. The strings' lengths range from 1 to 100 inclusive. It is guaranteed that the strings are of the same length and also consist of uppercase and lowercase Latin letters. Output If the first string is less than the second one, print "-1". If the second string is less than the first one, print "1". If the strings are equal, print "0". Note that the letters' case is not taken into consideration when the strings are compared. Examples Input aaaa aaaA Output 0 Input abs Abz Output -1 Input abcdefg AbCdEfF Output 1 Note If you want more formal information about the lexicographical order (also known as the "dictionary order" or "alphabetical order"), you can visit the following site: * http://en.wikipedia.org/wiki/Lexicographical_order The input will be give Read the inputs from stdin solve the problem and write the answer to stdout (do not directly test on the sample inputs). Enclose your code within ```python delimiters.

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