Initialization:

We need a function to calculate the sum of the array based on a given peak (maximum possible value at index) and the constraints.
Use binary search to determine the maximum possible value at index.

Binary Search:

Perform binary search on the possible values of nums[index] from 1 to maxSum.
For each candidate value, check if the constructed array satisfies the maxSum constraint.

Sum Calculation:

Determine the sum of the segment to the left of index, including index.
Determine the sum of the segment to the right of index.
Ensure the total sum does not exceed maxSum.

Code

[{"language":"Java","code":"public class Solution {\n    public int maxValue(int n, int index, int maxSum) {\n        int left = 1;\n        int right = maxSum;\n\n        while (left < right) {\n            int mid = (left + right + 1) / 2;\n            if (isValid(n, index, maxSum, mid)) {\n                left = mid;\n            } else {\n                right = mid - 1;\n            }\n        }\n\n        return left;\n    }\n\n    private boolean isValid(int n, int index, int maxSum, int value) {\n        long leftSum = sum(index, value - 1);\n        long rightSum = sum(n - index - 1, value - 1);\n        long totalSum = leftSum + rightSum + value;\n        return totalSum <= maxSum;\n    }\n\n    private long sum(int length, long value) {\n        if (length >= value) {\n            return (value * (value + 1)) / 2 + (length - value);\n        } else {\n            return (length * (2 * value - length + 1)) / 2;\n        }\n    }\n}","

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