Maximum Non-Adjacent Sum With Value Gap

You are given an integer array nums of non-negative values and a non-negative integer d.

Choose a set of indices, possibly empty, such that:

• no two chosen indices are adjacent (j != i + 1 for any two chosen i < j), and
• for every pair of chosen indices that are consecutive in the chosen set, meaning no other chosen index lies between them, the values differ by at least d: |nums[i] - nums[j]| >= d.

Return the maximum possible sum of the chosen values.

Examples

Example 1

Input: nums = [5, 1, 8, 2, 9], d = 3
Output: 14
Explanation: Choose indices 0 and 4 (values 5 and 9, difference 4 >= 3) for 14. Indices 0, 2, 4 are not allowed together because 8 and 9 differ by only 1.

Example 2

Input: nums = [4, 4, 4], d = 1
Output: 4
Explanation: Any two chosen values would be equal, so at most one index can be chosen.

Example 3

Input: nums = [10, 20, 30, 40], d = 15
Output: 60
Explanation: Indices 1 and 3 (values 20 and 40) differ by 20 >= 15.