Find max consecutive elements with sum below target

You are given:

• An integer array nums of length n, sorted in non-decreasing order.
• An integer index such that 0 ≤ index < n.
• An integer target.

Starting from position index, you may take a contiguous sequence of elements going to the right: nums[index], nums[index + 1], ..., nums[index + k - 1] for some integer k ≥ 0.

You want the sum of the chosen elements to be strictly less than target:

[ \sum_{i=index}^{index + k - 1} nums[i] < target. ]

Find the maximum possible value of k (the maximum number of consecutive elements starting from index whose sum is < target). If even nums[index] ≥ target, then the answer should be 0.

Design an efficient algorithm to compute this maximum k.

You may assume:

• 1 ≤ n
• nums is sorted in non-decreasing order.

Describe the algorithm you would implement and its time and space complexity.