<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:
        cache = {}
        def dp(i, mod):
            if i >= len(nums):
                return 0 if mod == 0 else float('-inf')
            if (i, mod) in cache:
                return cache[(i, mod)]
            
            # add_num = dp(i + 1, ((mod + 3) - (nums[i] % 3)) % 3)
            # dont_add_num = dp(i + 1, mod)
            res = max(nums[i] + dp(i + 1, ((mod + 3) - (nums[i] % 3)) % 3), dp(i + 1, mod)) # Either add this num or dont
            cache[(i, mod)] = res
            return res
        
        out = dp(0, 0) 
        print(cache)
        return out if out != float('-inf') else 0
            
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