<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:
    def solve_rec(self, index, arr, no_of_slices, end_index):
        if no_of_slices == 0 or index > end_index:
            return 0
        include = arr[index] + self.solve_rec(index + 2, arr, no_of_slices - 1, end_index)
        exclude = 0 + self.solve_rec(index + 1, arr, no_of_slices, end_index)
        return max(include, exclude)
    
    def solve_mem(self, index, arr, no_of_slices, end_index, dp):
        if no_of_slices == 0 or index > end_index:
            return 0
        if dp[index][no_of_slices] != -1:
            return dp[index][no_of_slices]
        include = arr[index] + self.solve_mem(index + 2, arr, no_of_slices - 1, end_index, dp)
        exclude  = 0 + self.solve_mem(index + 1, arr, no_of_slices, end_index, dp)
        dp[index][no_of_slices] = max(include, exclude)
        return dp[index][no_of_slices]
        
    def maxSizeSlices(self, slices: List[int]) -> int:
        k = len(slices)
        n = k // 3 # you can eat only k//3 slices
        # without_first = self.solve_rec(1, slices, n, k - 1)
        # without_last = self.solve_rec(0, slices, n, k - 2)
        dp1 = [[-1 for _ in range(n + 1)]for _ in range(k + 1)]
        without_first = self.solve_mem(1, slices, n, k - 1, dp1)
        dp2 = [[-1 for _ in range(n + 1)]for _ in range(k + 1)]
        without_last = self.solve_mem(0, slices, n, k - 2, dp2)
        return max(without_first, without_last)

        
20000