Maximize distinct values after unique ± offsets

## Problem You are given an integer array `a` of length `n` and a non-negative integer `k`. For each index `i`, you must choose: - an integer offset `d_i` such that `0 <= d_i <= k`, and - a sign, either `+` or `-`, and replace `a[i]` with `a[i] + d_i` or `a[i] - d_i`. **Constraint:** All chosen offsets must be **distinct** across indices (i.e., `d_i != d_j` for `i != j`). Your goal is to **maximize the number of distinct integers** in the resulting array after all replacements. Return that maximum possible number of distinct values. ## Example - Input: `a = [0, 0, 0]`, `k = 1` - Output: `3` - Explanation: Choose offsets `0, 1` (but offsets must be distinct, so for 3 elements we can use `d = 0, 1` only if `k=1`; instead, one valid construction is to allow `d` to be chosen per element distinctly within `[0,k]`—for this example, a maximal distinct outcome is `[-1, 0, 1]` by applying `-1, 0, +1` to the three zeros. ## Notes / Clarifications - Offsets are integers. - Offset `0` is allowed. - The sign choice is independent per element. ## Constraints (typical interview setting) - `1 <= n <= 2 * 10^5` - `0 <= k <= 10^9` - `-10^9 <= a[i] <= 10^9` Implement a function that returns the maximum possible number of distinct values.