<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:
        # 
        # 4 5 4 3 2 1  > 10
        # 3 4 3 2 1 0 > 10
        # 2 3 2 1 1 1 = 10 valid
        def valid(val):
            # 2 3 2 1 1 1
            # 3 * 6 - (1 + 1 + 2 + 3 + 4)

            # - (1 + 2 + ... + 4) inclusive
            # - (1 + 2 + 2 + 2)
            print(val)
            bounded_substract = 0
            right_i = n - index - 1
            max_right_substract = max(0, min(val - 1, right_i))
            if right_i > val - 1 and max_right_substract == val - 1:
                # bounded by positive fact
                # counts of consecutive this after arithmetic series till val - 1
                counts_right = right_i - val + 1
                print(f'added {(val - 1) * counts_right} cuz {val - 1} should remain {counts_right} times ')
                bounded_substract += (val - 1) * counts_right
           

            # -(1 + )
            left_i = index
            max_left_substract = max(0, min(val - 1, left_i))
            if max_left_substract == val - 1:
                # bounded by positive fact
                # counts of consecutive this after arithmetic series till val - 1
                counts_left = left_i - val + 1
                print(f'added {(val - 1) * counts_left} cuz {val - 1} should remain ')
                bounded_substract += (val - 1) * counts_left
            print(f'added {bounded_substract} to series till {max_left_substract} and {max_right_substract}')
            return (val * n - (max_right_substract + 1) * max_right_substract / 2 - (max_left_substract + 1) * max_left_substract / 2 - bounded_substract ) <= maxSum

        l = 0
        r = maxSum
        while l <= r:
            m = (l + r) // 2
            if valid(m):
                l = m + 1
            else:
                r = m - 1
        return r


            
                


         
        
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