<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:
        if sum(nums) % 3 == 0:
            return sum(nums)

        @cache
        def sums_at(i):
            if i == -1:
                return 0, 0, 0
            
            prev_0, prev_1, prev_2 = sums_at(i-1)

            r = nums[i] % 3
            r_0, r_1, r_2 = 0, 0, 0
            if r == 0:
                r_0 = prev_0 + nums[i]
                r_1 = prev_1 + nums[i] if prev_1 > 0 else 0
                r_2 = prev_2 + nums[i] if prev_2 > 0 else 0
            elif r == 1:
                r_0 = max(prev_0, prev_2 + nums[i] if prev_2 > 0 else 0)
                r_1 = max(prev_1, prev_0 + nums[i])
                r_2 = max(prev_2, prev_1 + nums[i] if prev_1 > 0 else 0)
            else:
                r_0 = max(prev_0, prev_1 + nums[i] if prev_1 > 0 else 0)
                r_1 = max(prev_1, prev_2 + nums[i] if prev_2 > 0 else 0)
                r_2 = max(prev_2, prev_0 + nums[i])
            
            return r_0, r_1, r_2
        
        max_sum = 0
        for i in range(len(nums)):
            max_sum, _, _ = sums_at(i)
        return max_sum
37500