Problem

Given an integer array nums and an integer k, define the k-radius average at index i as the average of the subarray centered at i with length 2k + 1:

• the subarray is nums[i-k ... i+k]
• if that subarray goes out of bounds, the value at i is -1

Return an array averages where averages[i] is the k-radius average of nums centered at index i, using integer division by the window length 2k + 1.

Notes

• The result at each valid position is the floor of the exact average for non-negative sums, and standard integer division behavior for integer arithmetic in the chosen language.
• Positions that cannot form a full centered window must be set to -1.

Input Format

• An integer array nums
• An integer k

Output Format

• Return an integer array of the same length as nums.
• For each index i, output the average of the centered window of size 2k+1 if it exists, otherwise -1.

Constraints

• 1 <= nums.length
• 0 <= k
• A centered window is valid only when i-k >= 0 and i+k < nums.length

Hints

• A naive solution recomputes the sum for every index and is too slow for large arrays.
• Precompute prefix sums so each window sum can be answered in constant time.