To Make 1

2000ms

524288K

Description:

There are $$$n$$$ positive integers written on the blackboard. Also, a positive number $$$k \geq 2$$$ is chosen, and none of the numbers on the blackboard are divisible by $$$k$$$. In one operation, you can choose any two integers $$$x$$$ and $$$y$$$, erase them and write one extra number $$$f(x + y)$$$, where $$$f(x)$$$ is equal to $$$x$$$ if $$$x$$$ is not divisible by $$$k$$$, otherwise $$$f(x) = f(x / k)$$$.

In the end, there will be a single number of the blackboard. Is it possible to make the final number equal to $$$1$$$? If so, restore any sequence of operations to do so.

Input:

The first line contains two integers $$$n$$$ and $$$k$$$ — the initial number of integers on the blackboard, and the chosen number ($$$2 \leq n \leq 16$$$, $$$2 \leq k \leq 2000$$$).

The second line contains $$$n$$$ positive integers $$$a_1, \ldots, a_n$$$ initially written on the blackboard. It is guaranteed that none of the numbers $$$a_i$$$ is divisible by $$$k$$$, and the sum of all $$$a_i$$$ does not exceed $$$2000$$$.

Output:

If it is impossible to obtain $$$1$$$ as the final number, print "NO" in the only line.

Otherwise, print "YES" on the first line, followed by $$$n - 1$$$ lines describing operations. The $$$i$$$-th of these lines has to contain two integers $$$x_i$$$ and $$$y_i$$$ to be erased and replaced with $$$f(x_i + y_i)$$$ on the $$$i$$$-th operation. If there are several suitable ways, output any of them.

Sample Input:

2 2
1 1

Sample Output:

YES
1 1

Sample Input:

4 3
7 8 13 23

Sample Output:

Sample Input:

3 4
1 2 3

Sample Output:

NO

Note:

In the second sample case:

$$$f(8 + 7) = f(15) = f(5) = 5$$$;
$$$f(23 + 13) = f(36) = f(12) = f(4) = 4$$$;
$$$f(5 + 4) = f(9) = f(3) = f(1) = 1$$$.

Informação

Provedor Codeforces

Código CF1225G