<p>Given <code>n</code> <code>cuboids</code> where the dimensions of the <code>i<sup>th</sup></code> cuboid is <code>cuboids[i] = [width<sub>i</sub>, length<sub>i</sub>, height<sub>i</sub>]</code> (<strong>0-indexed</strong>). Choose a <strong>subset</strong> of <code>cuboids</code> and place them on each other.</p>

<p>You can place cuboid <code>i</code> on cuboid <code>j</code> if <code>width<sub>i</sub> &lt;= width<sub>j</sub></code> and <code>length<sub>i</sub> &lt;= length<sub>j</sub></code> and <code>height<sub>i</sub> &lt;= height<sub>j</sub></code>. You can rearrange any cuboid&#39;s dimensions by rotating it to put it on another cuboid.</p>

<p>Return <em>the <strong>maximum height</strong> of the stacked</em> <code>cuboids</code>.</p>

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

<p><strong><img alt="" src="https://assets.leetcode.com/uploads/2019/10/21/image.jpg" style="width: 420px; height: 299px;" /></strong></p>

<pre>
<strong>Input:</strong> cuboids = [[50,45,20],[95,37,53],[45,23,12]]
<strong>Output:</strong> 190
<strong>Explanation:</strong>
Cuboid 1 is placed on the bottom with the 53x37 side facing down with height 95.
Cuboid 0 is placed next with the 45x20 side facing down with height 50.
Cuboid 2 is placed next with the 23x12 side facing down with height 45.
The total height is 95 + 50 + 45 = 190.
</pre>

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

<pre>
<strong>Input:</strong> cuboids = [[38,25,45],[76,35,3]]
<strong>Output:</strong> 76
<strong>Explanation:</strong>
You can&#39;t place any of the cuboids on the other.
We choose cuboid 1 and rotate it so that the 35x3 side is facing down and its height is 76.
</pre>

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

<pre>
<strong>Input:</strong> cuboids = [[7,11,17],[7,17,11],[11,7,17],[11,17,7],[17,7,11],[17,11,7]]
<strong>Output:</strong> 102
<strong>Explanation:</strong>
After rearranging the cuboids, you can see that all cuboids have the same dimension.
You can place the 11x7 side down on all cuboids so their heights are 17.
The maximum height of stacked cuboids is 6 * 17 = 102.
</pre>

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

<ul>
	<li><code>n == cuboids.length</code></li>
	<li><code>1 &lt;= n &lt;= 100</code></li>
	<li><code>1 &lt;= width<sub>i</sub>, length<sub>i</sub>, height<sub>i</sub> &lt;= 100</code></li>
</ul>

class Solution {
public:
int dfs(int currentIndex, const vector<vector<int>>& cuboids, vector<int>& dp) {
    // If already computed, return the stored value
    if (dp[currentIndex] != -1) {
        return dp[currentIndex];
    }
    
    int maxHeight = 0;
    // Try to place each cuboid on top of the current cuboid
    for (int i = currentIndex + 1; i < cuboids.size(); ++i) {
        if (cuboids[currentIndex][0] <= cuboids[i][0] &&
            cuboids[currentIndex][1] <= cuboids[i][1] &&
            cuboids[currentIndex][2] <= cuboids[i][2]) {
            maxHeight = max(maxHeight, dfs(i, cuboids, dp));
        }
    }

    // Include the current cuboid's height
    dp[currentIndex] = maxHeight + cuboids[currentIndex][2];
    return dp[currentIndex];
}

int maxHeight(vector<vector<int>>& cuboids) {
    int n = cuboids.size();

    // Step 1: Sort dimensions of each cuboid to allow easy comparison
    for (auto& cuboid : cuboids) {
        sort(cuboid.begin(), cuboid.end());
    }

    // Step 2: Sort cuboids based on width, length, and height
    sort(cuboids.begin(), cuboids.end());

    // Step 3: Memoization array
    vector<int> dp(n, -1);

    int maxHeight = 0;

    // Try to place each cuboid as the base cuboid
    for (int i = 0; i < n; ++i) {
        maxHeight = max(maxHeight, dfs(i, cuboids, dp));
    }

    return maxHeight;
}
};
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