"You may assume that all intermediate values of H and all values of F fit in range [0..100000]"

This should be put to problem statement, rather than to input description. Otherwise claim about periodicity is wrong.

Consider the following sequence:

X1=0, X2=2
A1=0, A2=0, A3=2, A4=0
B1=3, B2=2, C=5

0 2 -1 -2 -4 -8 -16 ... -(2^n) ...

It's not periodic, but all coefficients (as well as initial elements) are non-negative.