<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>

from typing import List


class Solution:
    def maxSumDivThree(self, nums: List[int]) -> int:
        """
        Given a list of integers, find the maximum sum of any subsequence that is divisible by 3.
        nums = [3, 6, 5, 1, 8]
        """

        n = len(nums)
        dp = [[0, 0, 0] for _ in range(n)]  # dp[i] keeps track of max sums mod 3

        # Initialize the first number in the dp array
        dp[0][nums[0] % 3] = nums[0]

        for i in range(1, n):
            # current_dp = dp[i - 1][:]  # Copy the previous state to avoid overwriting
            dp[i] = dp[i - 1][:]
            for pre_status in range(3):
                new_sum = dp[i - 1][pre_status] + nums[i]
                target_status = new_sum % 3
                dp[i][target_status] = max(dp[i][target_status], new_sum)

        print(f"dp: {dp}")
        return dp[n - 1][0]  # Return the maximum sum divisible by 3


# Example usage:
sol = Solution()
nums = [3, 6, 5, 1, 8]
print(sol.maxSumDivThree(nums))  # Output: 18

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