12556번 - De-RNG-ed (Large) 다국어

I want to make an online poker website. A very important component of such a system is the random number generator. It needs to be fast and random enough. Here is a compromise I came up with. I need a way to generate random numbers of length at most D. My plan is to select a prime number P ≤ 10^D. I am also going to pick non-negative integers A and B. Finally, I'm going to pick an integer seed S between 0 and P-1, inclusive.

To output my sequence of pseudo-random numbers, I'm going to first output S and then compute the new value of S like this:
S := (A*S + B) mod P.

Then I will output the new value of S as the next number in the sequence and update Sagain by using the same formula. I can repeat this as many times as I want.

Do you think that this is a good random number generator? Can you write a program that takes K consecutive elements of a sequence that was generated by my random number generator, and prints the next element of the sequence?

The first line of the input gives the number of test cases, T.  T test cases follow. Each one starts with a line containing D and K. The next line contains K consecutive elements generated by a random number generator of the kind described above.

Limits

• 1 ≤ T ≤ 100.
• 1 ≤ K ≤ 10.
• The K integers will be consecutive elements of a sequence generated by a random number generator of the type described above.
• 1 ≤ D ≤ 6.

For each test case, output one line containing "Case #x: y", where x is the case number (starting from 1) and y is either the next number in the sequence, or the string "I don't know." if the answer is ambiguous.