<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>

"""
target = 3, index = 7, n = 15
[1,1,1,1,1,1,2,3,2,1,1,1,1,1,1]
               |
"""
class Solution:
    def maxValue(self, n: int, index: int, maxSum: int) -> int:
        def get_sum(target):
            count = 0
            # left
            if target > index:
                count += ((target - index) + target) * (index + 1) // 2
            else:
                # contains multiple 1's
                count += (1 + target) * target // 2 + (index - target) + 1

            print(count)

            # right
            if n - index < target:
                count += (target + target - (n - index - 1)) * (n - index) // 2
            else:
                # contains multiple 1's
                count += (1 + target) * target // 2 + (n - index - target)

            print(count - target)
            return count - target

        def binary_search(left, right):
            while left <= right:
                mid = (left + right) // 2
                if get_sum(mid) <= maxSum:
                    left = mid + 1
                else:
                    right = mid - 1
            return right
            
        return binary_search(1, maxSum)
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