<p>You are given a 2D array of integers <code>envelopes</code> where <code>envelopes[i] = [w<sub>i</sub>, h<sub>i</sub>]</code> represents the width and the height of an envelope.</p>

<p>One envelope can fit into another if and only if both the width and height of one envelope are greater than the other envelope&#39;s width and height.</p>

<p>Return <em>the maximum number of envelopes you can Russian doll (i.e., put one inside the other)</em>.</p>

<p><strong>Note:</strong> You cannot rotate an envelope.</p>

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

<pre>
<strong>Input:</strong> envelopes = [[5,4],[6,4],[6,7],[2,3]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The maximum number of envelopes you can Russian doll is <code>3</code> ([2,3] =&gt; [5,4] =&gt; [6,7]).
</pre>

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

<pre>
<strong>Input:</strong> envelopes = [[1,1],[1,1],[1,1]]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>1 &lt;= envelopes.length &lt;= 10<sup>5</sup></code></li>
	<li><code>envelopes[i].length == 2</code></li>
	<li><code>1 &lt;= w<sub>i</sub>, h<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution {
private:
    bool binarySearch(int start, int end, vector<stack<int>>& stacks, int target) {
        if (start >= end) {
            return false;
        }
        int mid = (start + end) / 2;
        if (stacks[mid].top() == target) {
            return true;
        }
        if (stacks[mid].top() > target) {
            bool hasLeftCompetitor = binarySearch(start, mid, stacks, target);
            if (!hasLeftCompetitor) {
                stacks[mid].push(target);
            }
            return true;
        }
        return binarySearch(mid + 1, end, stacks, target);
    }
public:
    int maxEnvelopes(vector<vector<int>>& envelopes) {
        sort(
            envelopes.begin(), envelopes.end(), 
            [](vector<int>& a, vector<int>& b) {
                return a[0] < b[0] || (a[0] == b[0] && a[1] > b[1]);
            }
        );
        vector<stack<int>> stacks;
        for (vector<int>& envelope : envelopes) {
            bool hasPutOnStack = binarySearch(0, stacks.size(), stacks, envelope[1]);
            if (!hasPutOnStack) {
                stack<int> newS;
                newS.push(envelope[1]);
                stacks.push_back(newS);
            }
        }
        return stacks.size();
    }
};
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