<p>Given an integer array <code>nums</code>, return <em>the <strong>maximum possible sum </strong>of elements of the array such that it is divisible by three</em>.</p>

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

<pre>
<strong>Input:</strong> nums = [3,6,5,1,8]
<strong>Output:</strong> 18
<strong>Explanation:</strong> Pick numbers 3, 6, 1 and 8 their sum is 18 (maximum sum divisible by 3).</pre>

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

<pre>
<strong>Input:</strong> nums = [4]
<strong>Output:</strong> 0
<strong>Explanation:</strong> Since 4 is not divisible by 3, do not pick any number.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4,4]
<strong>Output:</strong> 12
<strong>Explanation:</strong> Pick numbers 1, 3, 4 and 4 their sum is 12 (maximum sum divisible by 3).
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 4 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    
    def maxSumDivThree(self, nums: List[int]) -> int:
        
        nums_len = len(nums)
        dp = []
        for i in range(nums_len):
            dp.append([-1 for _ in range(3)])

        def maximumSum(index, s):

            if index == nums_len:
                if s == 0:
                    return 0
                else:
                    return -1e10

            if dp[index][s] != -1:
                return dp[index][s]

            maxSum = max(maximumSum(index + 1, s), maximumSum(index + 1, (s + nums[index]) % 3) + nums[index])

            dp[index][s] = maxSum

            return maxSum

        return maximumSum(0, 0)

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