LCM Constraints

You are given $M$ triplets $(X_1, Y_1, B_1), (X_2, Y_2, B_2), \ldots, (X_M, Y_M, B_M)$. Find the number of sequences of positive integers $A_1, A_2, \ldots, A_N$ such that for each valid $i$, $\mathrm{lcm}(A_{X_i},A_{Y_i}) = B_i$, or determine that there is an infinite number of such sequences. Since the answer can be very large, compute it modulo $1,000,000,007$ ($10^9 + 7$). Note that the val