<p>You are given a <strong>0-indexed</strong> integer array <code>nums</code>.</p>

<p>A <strong>subsequence</strong> of <code>nums</code> having length <code>k</code> and consisting of <strong>indices</strong> <code>i<sub>0</sub>&nbsp;&lt;&nbsp;i<sub>1</sub> &lt;&nbsp;... &lt; i<sub>k-1</sub></code> is <strong>balanced</strong> if the following holds:</p>

<ul>
	<li><code>nums[i<sub>j</sub>] - nums[i<sub>j-1</sub>] &gt;= i<sub>j</sub> - i<sub>j-1</sub></code>, for every <code>j</code> in the range <code>[1, k - 1]</code>.</li>
</ul>

<p>A <strong>subsequence</strong> of <code>nums</code> having length <code>1</code> is considered balanced.</p>

<p>Return <em>an integer denoting the <strong>maximum</strong> possible <strong>sum of elements</strong> in a <strong>balanced</strong> subsequence of </em><code>nums</code>.</p>

<p>A <strong>subsequence</strong> of an array is a new <strong>non-empty</strong> array that is formed from the original array by deleting some (<strong>possibly none</strong>) of the elements without disturbing the relative positions of the remaining elements.</p>

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

<pre>
<strong>Input:</strong> nums = [3,3,5,6]
<strong>Output:</strong> 14
<strong>Explanation:</strong> In this example, the subsequence [3,5,6] consisting of indices 0, 2, and 3 can be selected.
nums[2] - nums[0] &gt;= 2 - 0.
nums[3] - nums[2] &gt;= 3 - 2.
Hence, it is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.
The subsequence consisting of indices 1, 2, and 3 is also valid.
It can be shown that it is not possible to get a balanced subsequence with a sum greater than 14.</pre>

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

<pre>
<strong>Input:</strong> nums = [5,-1,-3,8]
<strong>Output:</strong> 13
<strong>Explanation:</strong> In this example, the subsequence [5,8] consisting of indices 0 and 3 can be selected.
nums[3] - nums[0] &gt;= 3 - 0.
Hence, it is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.
It can be shown that it is not possible to get a balanced subsequence with a sum greater than 13.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [-2,-1]
<strong>Output:</strong> -1
<strong>Explanation:</strong> In this example, the subsequence [-1] can be selected.
It is a balanced subsequence, and its sum is the maximum among the balanced subsequences of nums.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 10<sup>5</sup></code></li>
	<li><code>-10<sup>9</sup> &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maxBalancedSubsequenceSum(self, nums: List[int]) -> int:
        def update(ind,val,l,r,seg,i):
            if l==r:
                seg[i]=val
                return
            mid=(l+r)//2
            if mid>=ind:
                update(ind,val,l,mid,seg,2*i+1)
            else:
                update(ind,val,mid+1,r,seg,2*i+2)
            seg[i]=max(seg[2*i+1],seg[2*i+2])
        
        def query(x,y,l,r,seg,i):
            if r<x or l>y:
                return 0
            if l>=x and r<=y:
                return seg[i]
            mid=(l+r)//2
            return max(query(x,y,l,mid,seg,2*i+1),query(x,y,mid+1,r,seg,2*i+2))
        
        val_ind=[]
        seg=[0]*len(nums)*4
        ind={}
        n=len(nums)
        for i in range(len(nums)):
            val_ind.append([nums[i],i,nums[i]-i])
        val_ind.sort(key=lambda x:x[2])

        for i in range(len(val_ind)):
            ind[val_ind[i][1]]=i

        for i in range(len(nums)-1,-1,-1):
            sort_ind=ind[i]
            max_result=query(sort_ind,n-1,0,n-1,seg,0)+nums[i]
            update(sort_ind,max_result,0,n-1,seg,0)
        return query(0,n-1,0,n-1,seg,0)

        



            
        
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