Deep in Busy Beaver's robotics lab sits an experimental Busy Beaver machine. Given a positive integer $$$X$$$ written in base $$$B$$$, it attempts to produce two positive integers $$$Y$$$ and $$$Z$$$ such that $$$X + Y = Z$$$, and the base-$$$B$$$ representations of $$$Y$$$ and $$$Z$$$ contain exactly the same multiset of digits.
The machine never produces leading zeroes and never outputs numbers with $$$2 \cdot 10^5$$$ or more digits. It has recently stopped functioning, so your task is to determine whether such $$$Y$$$ and $$$Z$$$ exist, and to output them if they do. You are given the digits of $$$X$$$ in base $$$B$$$ without leading zeroes.
The first line contains a single integer $$$T$$$ ($$$1 \le T \le 100$$$), the number of test cases.
Each test case consists of two lines. The first line of each test case contains two integers $$$N$$$ and $$$B$$$ ($$$1 \le N \le 10^5$$$; $$$2 \le B \le 10^5$$$).
The second line of each test case contains $$$N$$$ integers $$$a_1, a_2, \ldots, a_N$$$ ($$$0 \le a_i \le B-1$$$; $$$a_1 \ne 0$$$), representing the digits of $$$X$$$ in base $$$B$$$: $$$$$$ X = a_1 B^{N-1} + a_2 B^{N-2} + \cdots + a_N. $$$$$$
The sum of $$$N$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
For each test case, if no valid solution exists, output a single line containing NO.
Otherwise, output YES on the first line. On the second line output an integer $$$M$$$ ($$$1 \le M \le 2 \cdot 10^5$$$), the number of digits in both $$$Y$$$ and $$$Z$$$.
On the next line output $$$M$$$ digits $$$p_1, p_2, \ldots, p_M$$$ ($$$0 \le p_i \le B-1$$$; $$$p_1 \ne 0$$$), representing $$$$$$ Y = p_1 B^{M-1} + p_2 B^{M-2} + \cdots + p_M. $$$$$$ On the following line output $$$M$$$ digits $$$q_1, q_2, \ldots, q_M$$$ ($$$0 \le q_i \le B-1$$$; $$$q_1 \ne 0$$$), representing $$$$$$ Z = q_1 B^{M-1} + q_2 B^{M-2} + \cdots + q_M. $$$$$$ The digit sequences $$$(p_1, \ldots, p_M)$$$ and $$$(q_1, \ldots, q_M)$$$ must use exactly the same multiset of digits, and the integers they represent must satisfy $$$X + Y = Z$$$. You may print YES and NO in any mixture of uppercase and lowercase letters.
32 103 64 51 4 3 45 124 8 8 3 1
YES 2 1 5 5 1 YES 4 1 4 3 2 3 4 2 1 NO
In the first test case, with $$$B = 10$$$ and $$$X = 36$$$, a valid solution is $$$Y = 15$$$ and $$$Z = 51$$$, since $$$51 = 36 + 15$$$ and both numbers use the digits $$$\{1,5\}$$$.
In the second test case, with $$$B = 5$$$ and $$$X$$$ given by digits $$$(1, 4, 3, 4)_5$$$, $$$$$$ X = 1 \cdot 5^3 + 4 \cdot 5^2 + 3 \cdot 5^1 + 4 = 244. $$$$$$ One valid pair is $$$$$$ Y = (1, 4, 3, 2)_5 = 242, \qquad Z = (3, 4, 2, 1)_5 = 486, $$$$$$ which share the digit multiset $$$\{1,2,3,4\}$$$ and satisfy $$$X + Y = Z$$$.
