Find Products of Elements of Big Array

Define big_nums by concatenating, for each positive integer in order, the sorted list of powers of two that sum to it (its binary decomposition); big_nums therefore contains only powers of two. For each query [l,r,mod], compute the product of big_nums[l..r] mod mod — the core challenge is handling indices up to 1e15 by counting occurrences of each 2^k in a range and using modular exponentiation without constructing the array.