<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 solve(vector<vector<int>>& cuboids, int previousCuboid, int currentCuboid, int n, vector<vector<int>>& dp) {
    if (currentCuboid < 0) {
        return 0;
    }
    
    if (dp[previousCuboid + 1][currentCuboid] != -1) {
        return dp[previousCuboid + 1][currentCuboid];
    }
    
    // Option 1: Exclude the current cuboid
    int exclude = solve(cuboids, previousCuboid, currentCuboid - 1, n, dp);
    
    // Option 2: Include the current cuboid if it can be stacked on the previous one
    int include = 0;
    if (previousCuboid == -1 || 
        (cuboids[previousCuboid][0] >= cuboids[currentCuboid][0] && 
         cuboids[previousCuboid][1] >= cuboids[currentCuboid][1] && 
         cuboids[previousCuboid][2] >= cuboids[currentCuboid][2])) {
        
        include = cuboids[currentCuboid][2] + solve(cuboids, currentCuboid, currentCuboid - 1, n, dp);
    }
    
    // Memoize the result and return the maximum of both options
    return dp[previousCuboid + 1][currentCuboid] = max(include, exclude);
}

int maxHeight(vector<vector<int>>& cuboids) {
    // Sort dimensions of each cuboid to ensure they are in increasing order
    for (auto& cuboid : cuboids) {
        sort(cuboid.begin(), cuboid.end());
    }
    
    // Sort all cuboids based on their dimensions to facilitate stacking
    sort(cuboids.begin(), cuboids.end());
    
    int n = cuboids.size();
    // DP table initialized with -1
    vector<vector<int>> dp(n + 1, vector<int>(n, -1));
    
    // Start recursion with no previous cuboid (-1) and starting from the last cuboid
    return solve(cuboids, -1, n - 1, n, dp);
}

};
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