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.