<p>You are given three positive integers:&nbsp;<code>n</code>, <code>index</code>, and <code>maxSum</code>. You want to construct an array <code>nums</code> (<strong>0-indexed</strong>)<strong> </strong>that satisfies the following conditions:</p>

<ul>
	<li><code>nums.length == n</code></li>
	<li><code>nums[i]</code> is a <strong>positive</strong> integer where <code>0 &lt;= i &lt; n</code>.</li>
	<li><code>abs(nums[i] - nums[i+1]) &lt;= 1</code> where <code>0 &lt;= i &lt; n-1</code>.</li>
	<li>The sum of all the elements of <code>nums</code> does not exceed <code>maxSum</code>.</li>
	<li><code>nums[index]</code> is <strong>maximized</strong>.</li>
</ul>

<p>Return <code>nums[index]</code><em> of the constructed array</em>.</p>

<p>Note that <code>abs(x)</code> equals <code>x</code> if <code>x &gt;= 0</code>, and <code>-x</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 4, index = 2,  maxSum = 6
<strong>Output:</strong> 2
<strong>Explanation:</strong> nums = [1,2,<u><strong>2</strong></u>,1] is one array that satisfies all the conditions.
There are no arrays that satisfy all the conditions and have nums[2] == 3, so 2 is the maximum nums[2].
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 6, index = 1,  maxSum = 10
<strong>Output:</strong> 3
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= maxSum &lt;= 10<sup>9</sup></code></li>
	<li><code>0 &lt;= index &lt; n</code></li>
</ul>

class Solution:
    def maxValue(self, n: int, index: int, maxSum: int) -> int:
        def calc_sum(n):
            return n * (n + 1) // 2
        
        def check(target):
            # sum of nums[:index](1, 2, ..., target - 1)
            left_size = index
            if left_size >= target - 1:
                left_sum = calc_sum(target - 1) + (left_size - target + 1)
            else:
                left_sum = calc_sum(target - 1) - calc_sum(target - 1 - left_size)
            # sum of nums[index + 1:]
            right_size = n - index - 1
            if right_size >= target - 1:
                right_sum = calc_sum(target - 1) + (right_size - target + 1)
            else:
                right_sum = calc_sum(target - 1) - calc_sum(target - 1 - right_size)
            
            if left_sum + right_sum + target > maxSum:
                return False
            return True
        
        lo, hi = 1, maxSum + 1
        while lo < hi:
            mid = lo + (hi - lo) // 2
            if check(mid):
                lo = mid + 1
            else:
                hi = mid
        
        return lo - 1
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