meetforever/GRPO_CL huggingface dataset:
user: You're given two arrays $a[1 \dots n]$ and $b[1 \dots n]$, both of the same length $n$.

In order to perform a push operation, you have to choose three integers $l, r, k$ satisfying $1 \le l \le r \le n$ and $k > 0$. Then, you will add $k$ to elements $a_l, a_{l+1}, \ldots, a_r$.

For example, if $a = [3, 7, 1, 4, 1, 2]$ and you choose $(l = 3, r = 5, k = 2)$, the array $a$ will become $[3, 7, \underline{3, 6, 3}, 2]$.

You can do this operation at most once. Can you make array $a$ equal to array $b$?

(We consider that $a = b$ if and only if, for every $1 \le i \le n$, $a_i = b_i$)


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 20$) — the number of test cases in the input.

The first line of each test case contains a single integer $n$ ($1 \le n \le 100\ 000$) — the number of elements in each array.

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 1000$).

The third line of each test case contains $n$ integers $b_1, b_2, \ldots, b_n$ ($1 \le b_i \le 1000$).

It is guaranteed that the sum of $n$ over all test cases doesn't exceed $10^5$.


-----Output-----

For each test case, output one line containing "YES" if it's possible to make arrays $a$ and $b$ equal by performing at most once the described operation or "NO" if it's impossible.

You can print each letter in any case (upper or lower).


-----Example-----
Input
4
6
3 7 1 4 1 2
3 7 3 6 3 2
5
1 1 1 1 1
1 2 1 3 1
2
42 42
42 42
1
7
6

Output
YES
NO
YES
NO



-----Note-----

The first test case is described in the statement: we can perform a push operation with parameters $(l=3, r=5, k=2)$ to make $a$ equal to $b$.

In the second test case, we would need at least two operations to make $a$ equal to $b$.

In the third test case, arrays $a$ and $b$ are already equal.

In the fourth test case, it's impossible to make $a$ equal to $b$, because the integer $k$ has to be positive.
user: Watto, the owner of a spare parts store, has recently got an order for the mechanism that can process strings in a certain way. Initially the memory of the mechanism is filled with n strings. Then the mechanism should be able to process queries of the following type: "Given string s, determine if the memory of the mechanism contains string t that consists of the same number of characters as s and differs from s in exactly one position".

Watto has already compiled the mechanism, all that's left is to write a program for it and check it on the data consisting of n initial lines and m queries. He decided to entrust this job to you.


-----Input-----

The first line contains two non-negative numbers n and m (0 ≤ n ≤ 3·10^5, 0 ≤ m ≤ 3·10^5) — the number of the initial strings and the number of queries, respectively.

Next follow n non-empty strings that are uploaded to the memory of the mechanism.

Next follow m non-empty strings that are the queries to the mechanism.

The total length of lines in the input doesn't exceed 6·10^5. Each line consists only of letters 'a', 'b', 'c'.


-----Output-----

For each query print on a single line "YES" (without the quotes), if the memory of the mechanism contains the required string, otherwise print "NO" (without the quotes).


-----Examples-----
Input
2 3
aaaaa
acacaca
aabaa
ccacacc
caaac

Output
YES
NO
NO
user: Omkar is building a waterslide in his water park, and he needs your help to ensure that he does it as efficiently as possible.

Omkar currently has $n$ supports arranged in a line, the $i$-th of which has height $a_i$. Omkar wants to build his waterslide from the right to the left, so his supports must be nondecreasing in height in order to support the waterslide. In $1$ operation, Omkar can do the following: take any contiguous subsegment of supports which is nondecreasing by heights and add $1$ to each of their heights. 

Help Omkar find the minimum number of operations he needs to perform to make his supports able to support his waterslide!

An array $b$ is a subsegment of an array $c$ if $b$ can be obtained from $c$ by deletion of several (possibly zero or all) elements from the beginning and several (possibly zero or all) elements from the end.

An array $b_1, b_2, \dots, b_n$ is called nondecreasing if $b_i\le b_{i+1}$ for every $i$ from $1$ to $n-1$.


-----Input-----

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \leq t \leq 100$). Description of the test cases follows.

The first line of each test case contains an integer $n$ ($1 \leq n \leq 2 \cdot 10^5$) — the number of supports Omkar has.

The second line of each test case contains $n$ integers $a_{1},a_{2},...,a_{n}$ $(0 \leq a_{i} \leq 10^9)$ — the heights of the supports.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.


-----Output-----

For each test case, output a single integer — the minimum number of operations Omkar needs to perform to make his supports able to support his waterslide.


-----Example-----
Input
3
4
5 3 2 5
5
1 2 3 5 3
3
1 1 1

Output
3
2
0



-----Note-----

The subarray with which Omkar performs the operation is bolded.

In the first test case:

First operation:

$[5, 3, \textbf{2}, 5] \to [5, 3, \textbf{3}, 5]$

Second operation:

$[5, \textbf{3}, \textbf{3}, 5] \to [5, \textbf{4}, \textbf{4}, 5]$

Third operation:

$[5, \textbf{4}, \textbf{4}, 5] \to [5, \textbf{5}, \textbf{5}, 5]$

In the third test case, the array is already nondecreasing, so Omkar does $0$ operations.
user: You are given $n$ pairs of integers $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$. All of the integers in the pairs are distinct and are in the range from $1$ to $2 \cdot n$ inclusive.

Let's call a sequence of integers $x_1, x_2, \ldots, x_{2k}$ good if either   $x_1 < x_2 > x_3 < \ldots < x_{2k-2} > x_{2k-1} < x_{2k}$, or  $x_1 > x_2 < x_3 > \ldots > x_{2k-2} < x_{2k-1} > x_{2k}$. 

You need to choose a subset of distinct indices $i_1, i_2, \ldots, i_t$ and their order in a way that if you write down all numbers from the pairs in a single sequence (the sequence would be $a_{i_1}, b_{i_1}, a_{i_2}, b_{i_2}, \ldots, a_{i_t}, b_{i_t}$), this sequence is good.

What is the largest subset of indices you can choose? You also need to construct the corresponding index sequence $i_1, i_2, \ldots, i_t$.


-----Input-----

The first line contains single integer $n$ ($2 \leq n \leq 3 \cdot 10^5$) — the number of pairs.

Each of the next $n$ lines contain two numbers — $a_i$ and $b_i$ ($1 \le a_i, b_i \le 2 \cdot n$) — the elements of the pairs.

It is guaranteed that all integers in the pairs are distinct, that is, every integer from $1$ to $2 \cdot n$ is mentioned exactly once.


-----Output-----

In the first line print a single integer $t$ — the number of pairs in the answer.

Then print $t$ distinct integers $i_1, i_2, \ldots, i_t$ — the indexes of pairs in the corresponding order.


-----Examples-----
Input
5
1 7
6 4
2 10
9 8
3 5

Output
3
1 5 3

Input
3
5 4
3 2
6 1

Output
3
3 2 1



-----Note-----

The final sequence in the first example is $1 < 7 > 3 < 5 > 2 < 10$.

The final sequence in the second example is $6 > 1 < 3 > 2 < 5 > 4$.
user: You are given a directed acyclic graph with n vertices and m edges. There are no self-loops or multiple edges between any pair of vertices. Graph can be disconnected.

You should assign labels to all vertices in such a way that:

  Labels form a valid permutation of length n — an integer sequence such that each integer from 1 to n appears exactly once in it.  If there exists an edge from vertex v to vertex u then label_{v} should be smaller than label_{u}.  Permutation should be lexicographically smallest among all suitable. 

Find such sequence of labels to satisfy all the conditions.


-----Input-----

The first line contains two integer numbers n, m (2 ≤ n ≤ 10^5, 1 ≤ m ≤ 10^5).

Next m lines contain two integer numbers v and u (1 ≤ v, u ≤ n, v ≠ u) — edges of the graph. Edges are directed, graph doesn't contain loops or multiple edges.


-----Output-----

Print n numbers — lexicographically smallest correct permutation of labels of vertices.


-----Examples-----
Input
3 3
1 2
1 3
3 2

Output
1 3 2 

Input
4 5
3 1
4 1
2 3
3 4
2 4

Output
4 1 2 3 

Input
5 4
3 1
2 1
2 3
4 5

Output
3 1 2 4 5
user: You are given $n$ positive integers $a_1, \ldots, a_n$, and an integer $k \geq 2$. Count the number of pairs $i, j$ such that $1 \leq i < j \leq n$, and there exists an integer $x$ such that $a_i \cdot a_j = x^k$.


-----Input-----

The first line contains two integers $n$ and $k$ ($2 \leq n \leq 10^5$, $2 \leq k \leq 100$).

The second line contains $n$ integers $a_1, \ldots, a_n$ ($1 \leq a_i \leq 10^5$).


-----Output-----

Print a single integer — the number of suitable pairs.


-----Example-----
Input
6 3
1 3 9 8 24 1

Output
5



-----Note-----

In the sample case, the suitable pairs are: $a_1 \cdot a_4 = 8 = 2^3$; $a_1 \cdot a_6 = 1 = 1^3$; $a_2 \cdot a_3 = 27 = 3^3$; $a_3 \cdot a_5 = 216 = 6^3$; $a_4 \cdot a_6 = 8 = 2^3$.
user: Alice and Bob play a game. The game consists of several sets, and each set consists of several rounds. Each round is won either by Alice or by Bob, and the set ends when one of the players has won $x$ rounds in a row. For example, if Bob won five rounds in a row and $x = 2$, then two sets ends.

You know that Alice and Bob have already played $n$ rounds, and you know the results of some rounds. For each $x$ from $1$ to $n$, calculate the maximum possible number of sets that could have already finished if each set lasts until one of the players wins $x$ rounds in a row. It is possible that the last set is still not finished — in that case, you should not count it in the answer.


-----Input-----

The first line contains one integer $n$ ($1 \le n \le 10^6$) — the number of rounds.

The second line contains one string $s$ of length $n$ — the descriptions of rounds. If the $i$-th element of the string is 0, then Alice won the $i$-th round; if it is 1, then Bob won the $i$-th round, and if it is ?, then you don't know who won the $i$-th round.


-----Output-----

In the only line print $n$ integers. The $i$-th integer should be equal to the maximum possible number of sets that could have already finished if each set lasts until one of the players wins $i$ rounds in a row.


-----Examples-----
Input
6
11?000

Output
6 3 2 1 0 0 

Input
5
01?01

Output
5 1 0 0 0 

Input
12
???1??????1?

Output
12 6 4 3 2 2 1 1 1 1 1 1 



-----Note-----

Let's consider the first test case:

  if $x = 1$ and $s = 110000$ or $s = 111000$ then there are six finished sets;  if $x = 2$ and $s = 110000$ then there are three finished sets;  if $x = 3$ and $s = 111000$ then there are two finished sets;  if $x = 4$ and $s = 110000$ then there is one finished set;  if $x = 5$ then there are no finished sets;  if $x = 6$ then there are no finished sets.
user: Chubby Yang is studying linear equations right now. He came up with a nice problem. In the problem you are given an n × n matrix W, consisting of integers, and you should find two n × n matrices A and B, all the following conditions must hold:   A_{ij} = A_{ji}, for all i, j (1 ≤ i, j ≤ n);  B_{ij} =  - B_{ji}, for all i, j (1 ≤ i, j ≤ n);  W_{ij} = A_{ij} + B_{ij}, for all i, j (1 ≤ i, j ≤ n). 

Can you solve the problem?


-----Input-----

The first line contains an integer n (1 ≤ n ≤ 170). Each of the following n lines contains n integers. The j-th integer in the i-th line is W_{ij} (0 ≤ |W_{ij}| < 1717).


-----Output-----

The first n lines must contain matrix A. The next n lines must contain matrix B. Print the matrices in the format equal to format of matrix W in input. It is guaranteed that the answer exists. If there are multiple answers, you are allowed to print any of them.

The answer will be considered correct if the absolute or relative error doesn't exceed 10^{ - 4}.


-----Examples-----
Input
2
1 4
3 2

Output
1.00000000 3.50000000
3.50000000 2.00000000
0.00000000 0.50000000
-0.50000000 0.00000000

Input
3
1 2 3
4 5 6
7 8 9

Output
1.00000000 3.00000000 5.00000000
3.00000000 5.00000000 7.00000000
5.00000000 7.00000000 9.00000000
0.00000000 -1.00000000 -2.00000000
1.00000000 0.00000000 -1.00000000
2.00000000 1.00000000 0.00000000
user: Egor is a famous Russian singer, rapper, actor and blogger, and finally he decided to give a concert in the sunny Republic of Dagestan.

There are $n$ cities in the republic, some of them are connected by $m$ directed roads without any additional conditions. In other words, road system of Dagestan represents an arbitrary directed graph. Egor will arrive to the city $1$, travel to the city $n$ by roads along some path, give a concert and fly away.

As any famous artist, Egor has lots of haters and too annoying fans, so he can travel only by safe roads. There are two types of the roads in Dagestan, black and white: black roads are safe at night only, and white roads — in the morning. Before the trip Egor's manager's going to make a schedule: for each city he'll specify it's color, black or white, and then if during the trip they visit some city, the only time they can leave it is determined by the city's color: night, if it's black, and morning, if it's white. After creating the schedule Egor chooses an available path from $1$ to $n$, and for security reasons it has to be the shortest possible.

Egor's manager likes Dagestan very much and wants to stay here as long as possible, so he asks you to make such schedule that there would be no path from $1$ to $n$ or the shortest path's length would be greatest possible.

A path is one city or a sequence of roads such that for every road (excluding the first one) the city this road goes from is equal to the city previous road goes into. Egor can move only along paths consisting of safe roads only. 

The path length is equal to the number of roads in it. The shortest path in a graph is a path with smallest length.


-----Input-----

The first line contains two integers $n$, $m$ ($1 \leq n \leq 500000$, $0 \leq m \leq 500000$) — the number of cities and the number of roads.

The $i$-th of next $m$ lines contains three integers — $u_i$, $v_i$ and $t_i$ ($1 \leq u_i, v_i \leq n$, $t_i \in \{0, 1\}$) — numbers of cities connected by road and its type, respectively ($0$ — night road, $1$ — morning road).


-----Output-----

In the first line output the length of the desired path (or $-1$, if it's possible to choose such schedule that there's no path from $1$ to $n$).

In the second line output the desired schedule — a string of $n$ digits, where $i$-th digit is $0$, if the $i$-th city is a night one, and $1$ if it's a morning one.

If there are multiple answers, print any.


-----Examples-----
Input
3 4
1 2 0
1 3 1
2 3 0
2 3 1

Output
2
011
Input
4 8
1 1 0
1 3 0
1 3 1
3 2 0
2 1 0
3 4 1
2 4 0
2 4 1

Output
3
1101
Input
5 10
1 2 0
1 3 1
1 4 0
2 3 0
2 3 1
2 5 0
3 4 0
3 4 1
4 2 1
4 5 0

Output
-1
11111


-----Note-----

For the first sample, if we paint city $1$ white, the shortest path is $1 \rightarrow 3$. Otherwise, it's $1 \rightarrow 2 \rightarrow 3$ regardless of other cities' colors.

For the second sample, we should paint city $3$ black, and there are both black and white roads going from $2$ to $4$. Note that there can be a road connecting a city with itself.
user: You are given an array $a$ of length $2^n$. You should process $q$ queries on it. Each query has one of the following $4$ types:   $Replace(x, k)$ — change $a_x$ to $k$;  $Reverse(k)$ — reverse each subarray $[(i-1) \cdot 2^k+1, i \cdot 2^k]$ for all $i$ ($i \ge 1$);  $Swap(k)$ — swap subarrays $[(2i-2) \cdot 2^k+1, (2i-1) \cdot 2^k]$ and $[(2i-1) \cdot 2^k+1, 2i \cdot 2^k]$ for all $i$ ($i \ge 1$);  $Sum(l, r)$ — print the sum of the elements of subarray $[l, r]$. 

Write a program that can quickly process given queries.


-----Input-----

The first line contains two integers $n$, $q$ ($0 \le n \le 18$; $1 \le q \le 10^5$) — the length of array $a$ and the number of queries.

The second line contains $2^n$ integers $a_1, a_2, \ldots, a_{2^n}$ ($0 \le a_i \le 10^9$).

Next $q$ lines contains queries — one per line. Each query has one of $4$ types:   "$1$ $x$ $k$" ($1 \le x \le 2^n$; $0 \le k \le 10^9$) — $Replace(x, k)$;  "$2$ $k$" ($0 \le k \le n$) — $Reverse(k)$;  "$3$ $k$" ($0 \le k < n$) — $Swap(k)$;  "$4$ $l$ $r$" ($1 \le l \le r \le 2^n$) — $Sum(l, r)$. 

It is guaranteed that there is at least one $Sum$ query.


-----Output-----

Print the answer for each $Sum$ query.


-----Examples-----
Input
2 3
7 4 9 9
1 2 8
3 1
4 2 4

Output
24

Input
3 8
7 0 8 8 7 1 5 2
4 3 7
2 1
3 2
4 1 6
2 3
1 5 16
4 8 8
3 0

Output
29
22
1



-----Note-----

In the first sample, initially, the array $a$ is equal to $\{7,4,9,9\}$.

After processing the first query. the array $a$ becomes $\{7,8,9,9\}$.

After processing the second query, the array $a_i$ becomes $\{9,9,7,8\}$

Therefore, the answer to the third query is $9+7+8=24$.

In the second sample, initially, the array $a$ is equal to $\{7,0,8,8,7,1,5,2\}$. What happens next is:   $Sum(3, 7)$ $\to$ $8 + 8 + 7 + 1 + 5 = 29$;  $Reverse(1)$ $\to$ $\{0,7,8,8,1,7,2,5\}$;  $Swap(2)$ $\to$ $\{1,7,2,5,0,7,8,8\}$;  $Sum(1, 6)$ $\to$ $1 + 7 + 2 + 5 + 0 + 7 = 22$;  $Reverse(3)$ $\to$ $\{8,8,7,0,5,2,7,1\}$;  $Replace(5, 16)$ $\to$ $\{8,8,7,0,16,2,7,1\}$;  $Sum(8, 8)$ $\to$ $1$;  $Swap(0)$ $\to$ $\{8,8,0,7,2,16,1,7\}$.
user: You are given two integers $x$ and $y$. You can perform two types of operations:   Pay $a$ dollars and increase or decrease any of these integers by $1$. For example, if $x = 0$ and $y = 7$ there are four possible outcomes after this operation:   $x = 0$, $y = 6$;  $x = 0$, $y = 8$;  $x = -1$, $y = 7$;  $x = 1$, $y = 7$. 

  Pay $b$ dollars and increase or decrease both integers by $1$. For example, if $x = 0$ and $y = 7$ there are two possible outcomes after this operation:   $x = -1$, $y = 6$;  $x = 1$, $y = 8$.  

Your goal is to make both given integers equal zero simultaneously, i.e. $x = y = 0$. There are no other requirements. In particular, it is possible to move from $x=1$, $y=0$ to $x=y=0$.

Calculate the minimum amount of dollars you have to spend on it.


-----Input-----

The first line contains one integer $t$ ($1 \le t \le 100$) — the number of testcases.

The first line of each test case contains two integers $x$ and $y$ ($0 \le x, y \le 10^9$).

The second line of each test case contains two integers $a$ and $b$ ($1 \le a, b \le 10^9$).


-----Output-----

For each test case print one integer — the minimum amount of dollars you have to spend.


-----Example-----
Input
2
1 3
391 555
0 0
9 4

Output
1337
0



-----Note-----

In the first test case you can perform the following sequence of operations: first, second, first. This way you spend $391 + 555 + 391 = 1337$ dollars.

In the second test case both integers are equal to zero initially, so you dont' have to spend money.
user: Ivan has got an array of n non-negative integers a_1, a_2, ..., a_{n}. Ivan knows that the array is sorted in the non-decreasing order. 

Ivan wrote out integers 2^{a}_1, 2^{a}_2, ..., 2^{a}_{n} on a piece of paper. Now he wonders, what minimum number of integers of form 2^{b} (b ≥ 0) need to be added to the piece of paper so that the sum of all integers written on the paper equalled 2^{v} - 1 for some integer v (v ≥ 0). 

Help Ivan, find the required quantity of numbers.


-----Input-----

The first line contains integer n (1 ≤ n ≤ 10^5). The second input line contains n space-separated integers a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 2·10^9). It is guaranteed that a_1 ≤ a_2 ≤ ... ≤ a_{n}.


-----Output-----

Print a single integer — the answer to the problem.


-----Examples-----
Input
4
0 1 1 1

Output
0

Input
1
3

Output
3



-----Note-----

In the first sample you do not need to add anything, the sum of numbers already equals 2^3 - 1 = 7.

In the second sample you need to add numbers 2^0, 2^1, 2^2.
user: Dexterina and Womandark have been arch-rivals since they’ve known each other. Since both are super-intelligent teenage girls, they’ve always been trying to solve their disputes in a peaceful and nonviolent way. After god knows how many different challenges they’ve given to one another, their score is equal and they’re both desperately trying to best the other in various games of wits. This time, Dexterina challenged Womandark to a game of Nim.

Nim is a two-player game in which players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects from a single heap. The player who can't make a turn loses. By their agreement, the sizes of piles are selected randomly from the range [0, x]. Each pile's size is taken independently from the same probability distribution that is known before the start of the game.

Womandark is coming up with a brand new and evil idea on how to thwart Dexterina’s plans, so she hasn’t got much spare time. She, however, offered you some tips on looking fabulous in exchange for helping her win in Nim. Your task is to tell her what is the probability that the first player to play wins, given the rules as above.


-----Input-----

The first line of the input contains two integers n (1 ≤ n ≤ 10^9) and x (1 ≤ x ≤ 100) — the number of heaps and the maximum number of objects in a heap, respectively. The second line contains x + 1 real numbers, given with up to 6 decimal places each: P(0), P(1), ... , P(X). Here, P(i) is the probability of a heap having exactly i objects in start of a game. It's guaranteed that the sum of all P(i) is equal to 1.


-----Output-----

Output a single real number, the probability that the first player wins. The answer will be judged as correct if it differs from the correct answer by at most 10^{ - 6}.


-----Example-----
Input
2 2
0.500000 0.250000 0.250000

Output
0.62500000
user: Daniel has a string s, consisting of lowercase English letters and period signs (characters '.'). Let's define the operation of replacement as the following sequence of steps: find a substring ".." (two consecutive periods) in string s, of all occurrences of the substring let's choose the first one, and replace this substring with string ".". In other words, during the replacement operation, the first two consecutive periods are replaced by one. If string s contains no two consecutive periods, then nothing happens.

Let's define f(s) as the minimum number of operations of replacement to perform, so that the string does not have any two consecutive periods left.

You need to process m queries, the i-th results in that the character at position x_{i} (1 ≤ x_{i} ≤ n) of string s is assigned value c_{i}. After each operation you have to calculate and output the value of f(s).

Help Daniel to process all queries.


-----Input-----

The first line contains two integers n and m (1 ≤ n, m ≤ 300 000) the length of the string and the number of queries.

The second line contains string s, consisting of n lowercase English letters and period signs.

The following m lines contain the descriptions of queries. The i-th line contains integer x_{i} and c_{i} (1 ≤ x_{i} ≤ n, c_{i} — a lowercas English letter or a period sign), describing the query of assigning symbol c_{i} to position x_{i}.


-----Output-----

Print m numbers, one per line, the i-th of these numbers must be equal to the value of f(s) after performing the i-th assignment.


-----Examples-----
Input
10 3
.b..bz....
1 h
3 c
9 f

Output
4
3
1

Input
4 4
.cc.
2 .
3 .
2 a
1 a

Output
1
3
1
1



-----Note-----

Note to the first sample test (replaced periods are enclosed in square brackets).

The original string is ".b..bz....". after the first query f(hb..bz....) = 4    ("hb[..]bz...."  →  "hb.bz[..].."  →  "hb.bz[..]."  →  "hb.bz[..]"  →  "hb.bz.") after the second query f(hbс.bz....) = 3    ("hbс.bz[..].."  →  "hbс.bz[..]."  →  "hbс.bz[..]"  →  "hbс.bz.") after the third query f(hbс.bz..f.) = 1    ("hbс.bz[..]f."  →  "hbс.bz.f.")

Note to the second sample test.

The original string is ".cc.". after the first query: f(..c.) = 1    ("[..]c."  →  ".c.") after the second query: f(....) = 3    ("[..].."  →  "[..]."  →  "[..]"  →  ".") after the third query: f(.a..) = 1    (".a[..]"  →  ".a.") after the fourth query: f(aa..) = 1    ("aa[..]"  →  "aa.")