<p>You are given a <strong>0-indexed</strong> array of integers <code>nums</code>, and an integer <code>target</code>.</p>

<p>Return <em>the <strong>length of the longest subsequence</strong> of</em> <code>nums</code> <em>that sums up to</em> <code>target</code>. <em>If no such subsequence exists, return</em> <code>-1</code>.</p>

<p>A <strong>subsequence</strong> is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5], target = 9
<strong>Output:</strong> 3
<strong>Explanation:</strong> There are 3 subsequences with a sum equal to 9: [4,5], [1,3,5], and [2,3,4]. The longest subsequences are [1,3,5], and [2,3,4]. Hence, the answer is 3.
</pre>

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

<pre>
<strong>Input:</strong> nums = [4,1,3,2,1,5], target = 7
<strong>Output:</strong> 4
<strong>Explanation:</strong> There are 5 subsequences with a sum equal to 7: [4,3], [4,1,2], [4,2,1], [1,1,5], and [1,3,2,1]. The longest subsequence is [1,3,2,1]. Hence, the answer is 4.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,1,5,4,5], target = 3
<strong>Output:</strong> -1
<strong>Explanation:</strong> It can be shown that nums has no subsequence that sums up to 3.
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 1000</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 1000</code></li>
	<li><code>1 &lt;= target &lt;= 1000</code></li>
</ul>

class Solution:

    # Problem type: Dynamic programming (LCS).
    # Time complexity: O(nt).
    # Space compelxity: O(nt).
    def lengthOfLongestSubsequence(self, numbers: List[int], target: int) -> int:
        """Find the length of the longest subsequence that sums to the target."""
        # State: ik-th entry is the longest subsequence ending at index i that sums to k.
        # State initialization: no subsequences of length 0 sum to 0.
        table = [[-1] * (target + 1) for _ in range(len(numbers) + 1)]
        table[0][0] = 0

        # State exploration.
        for index in range(1, len(numbers) + 1):
            for value in range(target + 1):
                # State transition: we can either exclude the current element or
                # include it if it's less than the current target value.
                if table[index -1][value] != -1:
                    table[index][value] = max(
                        table[index][value],
                        table[index - 1][value],
                    )
                if numbers[index - 1] <= value and table[index - 1][value - numbers[index - 1]] != -1:
                    table[index][value] = max(
                        table[index][value],
                        table[index - 1][value - numbers[index - 1]] + 1,
                )
        return table[len(numbers)][target]
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