CF1225G.To Make 1

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.

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 $2000$ .

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.

In the second sample case:

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