<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 FenwickTree:
  def __init__(self, n: int):
    self.vals = [0] * (n + 1)

  def maximize(self, i: int, val: int) -> None:
    """Updates the maximum sum of subsequence ending in (i - 1) with `val`."""
    while i < len(self.vals):
      self.vals[i] = max(self.vals[i], val)
      i += FenwickTree.lowbit(i)

  def get(self, i: int) -> int:
    """Returns the maximum sum of subsequence ending in (i - 1)."""
    res = 0
    while i > 0:
      res = max(res, self.vals[i])
      i -= FenwickTree.lowbit(i)
    return res

  @staticmethod
  def lowbit(i: int) -> int:
    return i & -i


class Solution:
  def maxBalancedSubsequenceSum(self, nums: list[int]) -> int:
    # Let's define maxSum[i] := subsequence with the maximum sum ending in i
    # By observation:
    #    nums[i] - nums[j] >= i - j
    # => nums[i] - i >= nums[j] - j
    # So, if nums[i] - i >= nums[j] - j, where i > j,
    # maxSum[i] = max(maxSum[i], maxSum[j] + nums[i])
    ans = -math.inf
    tree = FenwickTree(len(nums))

    for _, i in sorted([(num - i, i) for i, num in enumerate(nums)]):
      subseqSum = tree.get(i) + nums[i]
      tree.maximize(i + 1, subseqSum)
      ans = max(ans, subseqSum)

    return ans
41800