You are given a **sorted** array of integers `nums` in non-decreasing order, and an integer `k ≥ 2`. Every value in `nums` appears **exactly `k` times in a row**, except for a single value that appears **exactly once**. Your task is to find and return that single value. Assume the length of the array is large, and you should aim for an efficient solution. We define a function: ```python find_single_with_k(nums, k) -> int ``` that returns the unique element. **Examples** ```text Example 1: Input: nums = [1, 1, 1, 2, 2, 2, 4, 5, 5, 5], k = 3 Output: 4 Explanation: - 1 appears 3 times, - 2 appears 3 times, - 5 appears 3 times, - 4 appears exactly once ⇒ answer = 4. Example 2: Input: nums = [1, 1, 1, 1, 2, 2, 2, 2, 4, 5, 5, 5, 5], k = 4 Output: 4 Explanation: - 1, 2, and 5 each appear 4 times, - 4 appears exactly once ⇒ answer = 4. ``` You may assume: - `nums.length >= 1`. - All integers that are not the unique one appear exactly `k` times. - Exactly one integer appears only once. --- **Follow-up 1:** Is there a relation between the array length `n = len(nums)` and the repetition count `k`? Express `n` in terms of `k` and the number of repeated groups. (For the intended setup, the correct relation is of the form: - `n = k * m + 1` for some integer `m ≥ 1`.) Explain why this must be true. --- **Follow-up 2 (binary search):** The array is sorted and organized as groups of size `k` (except for the single element). Use this structure to design an **O(log n)** time algorithm (better than O(n)) to find the single element. Describe how to apply **binary search** over indices to locate the position of the single element, using the grouping pattern implied by `k`.