<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:
        #max at index and taper off
        #binary search it
        #10 9 8 7
        #10 10 10 10
        #6
        #34
        def can_make(x):
            if x == 0:
                return True
            num_right = min(x, n - index - 1)
            num_left = min(x, index)
            left = num_left * (x) - ((num_left) * (num_left + 1) / 2)
            right = num_right * (x) - ((num_right) * (num_right + 1) / 2)
            print("inside", x, left, right, num_left)   
            print("inside", x, left, right, num_right)
            #how do i know whether to add ones
            if (n - 1 - index) > x - 1:
                print("need ones right", right, (n - 1 - index) - (x) + 1)
                right += (n - 1 - index) - (x) + 1
            if index >= x - 1:
                print("need ones left", left, index - x + 1)
                left += index - x + 1
            if left + right + x <= maxSum:
                return True
            return False
        l = 0
        h = maxSum
        while l <= h:
            m = (l + h) // 2
            print(m, can_make(m))
            if can_make(m):
                l = m + 1
            else:
                h = m - 1
        return h

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