Compute distinct sums from limited coins

You are given two integer arrays, denom[0..n-1] and count[0..n-1], where denom[i] > 0 is a coin denomination and count[i] >= 0 is the number of available coins of that denomination. Return the number of distinct positive sums that can be formed using at most count[i] coins of value denom[i] (each coin is identical and can be used at most once). Describe your algorithm and analyze its time and space complexity. Discuss how you would handle large maximum reachable sums (e.g., up to 1e 6).