Problem Statement

You are given a string S of length N. Let S_i\ (1\leq i \leq N) be the i-th character of S from the left.

You may perform the following two kinds of operations zero or more times in any order:

• Pay A yen (the currency in Japan). Move the leftmost character of S to the right end. In other words, change S_1S_2\ldots S_N to S_2\ldots S_NS_1.

• Pay B yen. Choose an integer i between 1 and N, and replace S_i with any lowercase English letter.

How many yen do you need to pay to make S a palindrome?

Constraints

• 1\leq N \leq 5000
• 1\leq A,B\leq 10^9
• S is a string of length N consisting of lowercase English letters.
• All values in the input except for S are integers.

Sample Input 1

5 1 2
rrefa

Sample Output 1

3

First, pay 2 yen to perform the operation of the second kind once: let i=5 to replace S_5 with e. S is now rrefe.

Then, pay 1 yen to perform the operation of the first kind once. S is now refer, which is a palindrome.

Thus, you can make S a palindrome for 3 yen. Since you cannot make S a palindrome for 2 yen or less, 3 is the answer.

Sample Input 2

8 1000000000 1000000000
bcdfcgaa

Sample Output 2

4000000000

Note that the answer may not fit into a 32-bit integer type.

Problem Statement

This problem is an easier version of Problem G.

There is a slot machine with three reels.
The arrangement of symbols on the i-th reel is represented by the string S_i. Here, S_i is a string of length M consisting of digits.

Each reel has a corresponding button. For each non-negative integer t, Takahashi can either choose and press one button or do nothing exactly t seconds after the reels start spinning.
If he presses the button corresponding to the i-th reel exactly t seconds after the reels start spinning, the i-th reel will stop and display the ((t \bmod M)+1)-th character of S_i.
Here, t \bmod M denotes the remainder when t is divided by M.

Takahashi wants to stop all the reels so that all the displayed characters are the same.
Find the minimum possible number of seconds from the start of the spin until all the reels are stopped so that his goal is achieved.
If this is impossible, report that fact.

Constraints

• 1 \leq M \leq 100
• M is an integer.
• S_i is a string of length M consisting of digits.

Sample Input 1

10
1937458062
8124690357
2385760149

Sample Output 1

6

Takahashi can stop each reel as follows so that 6 seconds after the reels start spinning, all the reels display 8.

• Press the button corresponding to the second reel 0 seconds after the reels start spinning. The second reel stops and displays 8, the ((0 \bmod 10)+1=1)-st character of S_2.
• Press the button corresponding to the third reel 2 seconds after the reels start spinning. The third reel stops and displays 8, the ((2 \bmod 10)+1=3)-rd character of S_3.
• Press the button corresponding to the first reel 6 seconds after the reels start spinning. The first reel stops and displays 8, the ((6 \bmod 10)+1=7)-th character of S_1.

There is no way to make the reels display the same character in 5 or fewer seconds, so print 6.

Sample Input 2

20
01234567890123456789
01234567890123456789
01234567890123456789

Sample Output 2

20

Note that he must stop all the reels and make them display the same character.

Sample Input 3

5
11111
22222
33333

Sample Output 3

-1

It is impossible to stop the reels so that all the displayed characters are the same.
In this case, print -1.

Problem Statement

The AtCoder Archipelago consists of N islands connected by N bridges. The islands are numbered from 1 to N, and the i-th bridge (1\leq i\leq N-1) connects islands i and i+1 bidirectionally, while the N-th bridge connects islands N and 1 bidirectionally. There is no way to travel between islands other than crossing the bridges.

On the islands, a tour that starts from island X_1 and visits islands X_2, X_3, \dots, X_M in order is regularly conducted. The tour may pass through islands other than those being visited, and the total number of times bridges are crossed during the tour is defined as the length of the tour.

More precisely, a tour is a sequence of l+1 islands a_0, a_1, \dots, a_l that satisfies all the following conditions, and its length is defined as l:

• For all j\ (0\leq j\leq l-1), islands a_j and a_{j+1} are directly connected by a bridge.
• There are some 0 = y_1 < y_2 < \dots < y_M = l such that for all k\ (1\leq k\leq M), a_{y_k} = X_k.

Due to financial difficulties, the islands will close one bridge to reduce maintenance costs. Determine the minimum possible length of the tour when the bridge to be closed is chosen optimally.

Constraints

• 3\leq N \leq 2\times 10^5
• 2\leq M \leq 2\times 10^5
• 1\leq X_k\leq N
• X_k\neq X_{k+1}\ (1\leq k\leq M-1)
• All input values are integers.

Sample Input 1

3 3
1 3 2

Sample Output 1

2

• If the first bridge is closed: By taking the sequence of islands (a_0, a_1, a_2) = (1, 3, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 2 can be conducted. There is no shorter tour.
• If the second bridge is closed: By taking the sequence of islands (a_0, a_1, a_2, a_3) = (1, 3, 1, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 3 can be conducted. There is no shorter tour.
• If the third bridge is closed: By taking the sequence of islands (a_0, a_1, a_2, a_3) = (1, 2, 3, 2), it is possible to visit islands 1, 3, 2 in order, and a tour of length 3 can be conducted. There is no shorter tour.

Therefore, the minimum possible length of the tour when the bridge to be closed is chosen optimally is 2.

The following figure shows, from left to right, the cases when bridges 1, 2, 3 are closed, respectively. The circles with numbers represent islands, the lines connecting the circles represent bridges, and the blue arrows represent the shortest tour routes.

Sample Input 2

4 5
2 4 2 4 2

Sample Output 2

8

The same island may appear multiple times in X_1, X_2, \dots, X_M.

Sample Input 3

163054 10
62874 19143 77750 111403 29327 56303 6659 18896 64175 26369

Sample Output 3

390009

Problem Statement

You are given sequences of length N: A = (A_1, A_2, \dots, A_N) and B = (B_1, B_2, \dots, B_N).
Let S be a subset of \lbrace1, 2, \dots, N\rbrace of size K. Here, find the minimum possible value of the following expression:

You are given T test cases; solve each of them.

Constraints

• 1 \leq T \leq 2 \times 10^5
• 1 \leq K \leq N \leq 2 \times 10^5
• 1 \leq A_i, B_i \leq 10^6
• The sum of N over all test cases is at most 2 \times 10^5.
• All input values are integers.

Sample Input 1

3
3 2
3 7 6
9 2 4
5 3
6 4 1 5 9
8 6 5 1 7
10 6
61 95 61 57 69 49 46 47 14 43
39 79 48 92 90 76 30 16 30 94

Sample Output 1

42
60
14579

In the first test case, for S = \{2, 3\}, the value of the expression is 7 \times (2 + 4) = 42, which is the minimum.

Problem Statement

You are given three integer sequences of length N, namely A=(A_1,A_2,\ldots,A_N), B=(B_1,B_2,\ldots,B_N), and C=(C_1,C_2,\ldots,C_N), and an integer K.

For each of the N^3 choices of integers i,j,k (1\leq i,j,k\leq N), compute the value A_iB_j + B_jC_k + C_kA_i. Among all these values, find the K-th largest value.

Constraints

• 1\leq N \leq 2\times 10^5
• 1\leq K \leq \min(N^3,5\times 10^5)
• 1\leq A_i,B_i,C_i \leq 10^9
• All input values are integers.

Sample Input 1

2 5
1 2
3 4
5 6

Sample Output 1

31

The N^3=8 values are computed as follows:

• For (i,j,k)=(1,1,1): A_1B_1+B_1C_1+C_1A_1=1\times 3+3\times 5+5\times 1=23
• For (i,j,k)=(1,1,2): A_1B_1+B_1C_2+C_2A_1=1\times 3+3\times 6+6\times 1=27
• For (i,j,k)=(1,2,1): A_1B_2+B_2C_1+C_1A_1=1\times 4+4\times 5+5\times 1=29
• For (i,j,k)=(1,2,2): A_1B_2+B_2C_2+C_2A_1=1\times 4+4\times 6+6\times 1=34
• For (i,j,k)=(2,1,1): A_2B_1+B_1C_1+C_1A_2=2\times 3+3\times 5+5\times 2=31
• For (i,j,k)=(2,1,2): A_2B_1+B_1C_2+C_2A_2=2\times 3+3\times 6+6\times 2=36
• For (i,j,k)=(2,2,1): A_2B_2+B_2C_1+C_1A_2=2\times 4+4\times 5+5\times 2=38
• For (i,j,k)=(2,2,2): A_2B_2+B_2C_2+C_2A_2=2\times 4+4\times 6+6\times 2=44

Sorting these values in descending order, we have (44,38,36,34,31,29,27,23), so the 5th largest value is 31.

Sample Input 2

3 10
100 100 100
100 100 100
100 100 100

Sample Output 2

30000

Sample Input 3

5 54
800516877 573289179 26509423 168629803 696409999
656737335 915059758 201458890 931198638 185928366
140174496 254538849 830992027 305186313 322164559

Sample Output 3

689589940713840351