<p>There is a pizza with <code>3n</code> slices of varying size, you and your friends will take slices of pizza as follows:</p>

<ul>
	<li>You will pick <strong>any</strong> pizza slice.</li>
	<li>Your friend Alice will pick the next slice in the anti-clockwise direction of your pick.</li>
	<li>Your friend Bob will pick the next slice in the clockwise direction of your pick.</li>
	<li>Repeat until there are no more slices of pizzas.</li>
</ul>

<p>Given an integer array <code>slices</code> that represent the sizes of the pizza slices in a clockwise direction, return <em>the maximum possible sum of slice sizes that you can pick</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/18/sample_3_1723.png" style="width: 500px; height: 266px;" />
<pre>
<strong>Input:</strong> slices = [1,2,3,4,5,6]
<strong>Output:</strong> 10
<strong>Explanation:</strong> Pick pizza slice of size 4, Alice and Bob will pick slices with size 3 and 5 respectively. Then Pick slices with size 6, finally Alice and Bob will pick slice of size 2 and 1 respectively. Total = 4 + 6.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/02/18/sample_4_1723.png" style="width: 500px; height: 299px;" />
<pre>
<strong>Input:</strong> slices = [8,9,8,6,1,1]
<strong>Output:</strong> 16
<strong>Explanation:</strong> Pick pizza slice of size 8 in each turn. If you pick slice with size 9 your partners will pick slices of size 8.
</pre>

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

<ul>
	<li><code>3 * n == slices.length</code></li>
	<li><code>1 &lt;= slices.length &lt;= 500</code></li>
	<li><code>1 &lt;= slices[i] &lt;= 1000</code></li>
</ul>

class Solution {
public:

    int solve(int index, int endIndex, vector<int>& slices, int n){
        //base case
        if(n==0 || index > endIndex){
            return 0;

        }

        int take = slices[index] + solve(index+2, endIndex, slices, n-1);
        int notTake = 0 +  solve(index+1, endIndex, slices, n);

        return max(take, notTake);
    }

    int solveMemo(int index, int endIndex, vector<int>& slices, int n, vector<vector<int>> dp){
        //base case
        if(n==0 || index > endIndex){
            return 0;

        }

        if(dp[index][n] != -1) return dp[index][n]; 

        int take = slices[index] + solveMemo(index+2, endIndex, slices, n-1, dp);
        int notTake = 0 +  solveMemo(index+1, endIndex, slices, n, dp);

        return dp[index][n] = max(take, notTake);
    }

    int solveTab(vector<int>& slices){
        int k = slices.size();

        vector<vector<int>> dp1(k+2, vector<int> (k+2, 0));
        vector<vector<int>> dp2(k+2, vector<int> (k+2, 0));

        for(int index = k-2; index >= 0; index--){
            for(int n = 1; n<=k/3; n++){
                int take = slices[index] + dp1[index+2][n-1];
                int notTake = 0 + dp1[index+1][n];

                dp1[index][n] = max(take, notTake);
            }
        }

        int case1 = dp1[0][k/3];

        for(int index = k-1; index >= 1; index--){
            for(int n = 1; n<=k/3; n++){
                int take = slices[index] + dp2[index+2][n-1];
                int notTake = 0 + dp2[index+1][n];

                dp2[index][n] = max(take, notTake);
            }
        }

        int case2 = dp2[1][k/3];

        return max(case1, case2);

    }

    int maxSizeSlices(vector<int>& slices) {
        int k = slices.size();

        // int case1 = solve(0, k-2, slices, k/3);
        // int case2 = solve(1, k-1, slices, k/3);

        // return max(case1, case2);

        // vector<vector<int>> dp1(k, vector<int> (k, -1));
        // vector<vector<int>> dp2(k, vector<int> (k, -1));

        // int case1 = solveMemo(0, k-2, slices, k/3, dp1);
        // int case2 = solveMemo(1, k-1, slices, k/3, dp2);

        // return max(case1, case2);

        return solveTab( slices);
        
    }


};
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