Maximum Product After K Increments

Statement

You are given an array, nums, consisting of non-negative integers, and an integer k representing the maximum number of allowed operations.

In each operation, you may select any element in nums and increment it by $1$. You can perform, at most, k such operations.

Your task is to maximize the product of all elements in the array after performing up to k operations. As the resulting product can be very large, return the product modulo $10^9$ ...