A. Anonymous Informant
time limit per test
3 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

You are given an array $$$b_1, b_2, \ldots, b_n$$$.

An anonymous informant has told you that the array $$$b$$$ was obtained as follows: initially, there existed an array $$$a_1, a_2, \ldots, a_n$$$, after which the following two-component operation was performed $$$k$$$ times:

  1. A fixed point$$$^{\dagger}$$$ $$$x$$$ of the array $$$a$$$ was chosen.
  2. Then, the array $$$a$$$ was cyclically shifted to the left$$$^{\ddagger}$$$ exactly $$$x$$$ times.

As a result of $$$k$$$ such operations, the array $$$b_1, b_2, \ldots, b_n$$$ was obtained. You want to check if the words of the anonymous informant can be true or if they are guaranteed to be false.

$$$^{\dagger}$$$A number $$$x$$$ is called a fixed point of the array $$$a_1, a_2, \ldots, a_n$$$ if $$$1 \leq x \leq n$$$ and $$$a_x = x$$$.

$$$^{\ddagger}$$$A cyclic left shift of the array $$$a_1, a_2, \ldots, a_n$$$ is the array $$$a_2, \ldots, a_n, a_1$$$.

Input

Each test contains multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $$$n, k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le k \le 10^9$$$) — the length of the array $$$b$$$ and the number of operations performed.

The second line of each test case contains $$$n$$$ integers $$$b_1, b_2, \ldots, b_n$$$ ($$$1 \le b_i \le 10^9$$$) — the elements of the array $$$b$$$.

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

Output

For each test case, output "Yes" if the words of the anonymous informant can be true, and "No" if they are guaranteed to be false.

Example
Input
6
5 3
4 3 3 2 3
3 100
7 2 1
5 5
6 1 1 1 1
1 1000000000
1
8 48
9 10 11 12 13 14 15 8
2 1
1 42
Output
Yes
Yes
No
Yes
Yes
No
Note

In the first test case, the array $$$a$$$ could be equal to $$$[3, 2, 3, 4, 3]$$$. In the first operation, a fixed point $$$x = 2$$$ was chosen, and after $$$2$$$ left shifts, the array became $$$[3, 4, 3, 3, 2]$$$. In the second operation, a fixed point $$$x = 3$$$ was chosen, and after $$$3$$$ left shifts, the array became $$$[3, 2, 3, 4, 3]$$$. In the third operation, a fixed point $$$x = 3$$$ was chosen again, and after $$$3$$$ left shifts, the array became $$$[4, 3, 3, 2, 3]$$$, which is equal to the array $$$b$$$.

In the second test case, the array $$$a$$$ could be equal to $$$[7, 2, 1]$$$. After the operation with a fixed point $$$x = 2$$$, the array became $$$[1, 7, 2]$$$. Then, after the operation with a fixed point $$$x = 1$$$, the array returned to its initial state $$$[7, 2, 1]$$$. These same $$$2$$$ operations (with $$$x = 2$$$, and $$$x = 1$$$) were repeated $$$49$$$ times. So, after $$$100$$$ operations, the array returned to $$$[7, 2, 1]$$$.

In the third test case, it can be shown that there is no solution.