<p>You are given an array of strings <code>words</code> and a string <code>target</code>.</p>

<p>A string <code>x</code> is called <strong>valid</strong> if <code>x</code> is a <span data-keyword="string-prefix">prefix</span> of <strong>any</strong> string in <code>words</code>.</p>

<p>Return the <strong>minimum</strong> number of <strong>valid</strong> strings that can be <em>concatenated</em> to form <code>target</code>. If it is <strong>not</strong> possible to form <code>target</code>, return <code>-1</code>.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abc&quot;,&quot;aaaaa&quot;,&quot;bcdef&quot;], target = &quot;aabcdabc&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">3</span></p>

<p><strong>Explanation:</strong></p>

<p>The target string can be formed by concatenating:</p>

<ul>
	<li>Prefix of length 2 of <code>words[1]</code>, i.e. <code>&quot;aa&quot;</code>.</li>
	<li>Prefix of length 3 of <code>words[2]</code>, i.e. <code>&quot;bcd&quot;</code>.</li>
	<li>Prefix of length 3 of <code>words[0]</code>, i.e. <code>&quot;abc&quot;</code>.</li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abababab&quot;,&quot;ab&quot;], target = &quot;ababaababa&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">2</span></p>

<p><strong>Explanation:</strong></p>

<p>The target string can be formed by concatenating:</p>

<ul>
	<li>Prefix of length 5 of <code>words[0]</code>, i.e. <code>&quot;ababa&quot;</code>.</li>
	<li>Prefix of length 5 of <code>words[0]</code>, i.e. <code>&quot;ababa&quot;</code>.</li>
</ul>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">words = [&quot;abcdef&quot;], target = &quot;xyz&quot;</span></p>

<p><strong>Output:</strong> <span class="example-io">-1</span></p>
</div>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 100</code></li>
	<li><code>1 &lt;= words[i].length &lt;= 5 * 10<sup>4</sup></code></li>
	<li>The input is generated such that <code>sum(words[i].length) &lt;= 10<sup>5</sup></code>.</li>
	<li><code>words[i]</code> consists only of lowercase English letters.</li>
	<li><code>1 &lt;= target.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>target</code> consists only of lowercase English letters.</li>
</ul>

from typing import List

class Solution:
    def minValidStrings(self, words: List[str], target: str) -> int:
        N = len(target)
        matches = [[] for _ in range(N)]  # matches[i] will store all prefix lengths matching starting at i

        for word in words:
            len_word = len(word)
            if len_word == 0:
                continue  # Skip empty words

            # Build the prefix function for the current word
            prefix = [0] * len_word
            for i in range(1, len_word):
                j = prefix[i - 1]
                while j > 0 and word[i] != word[j]:
                    j = prefix[j - 1]
                if word[i] == word[j]:
                    j += 1
                prefix[i] = j

            # Perform KMP search of the word in the target
            j = 0  # Index for word
            for i in range(N):
                while j > 0 and target[i] != word[j]:
                    j = prefix[j - 1]
                if target[i] == word[j]:
                    j += 1
                # Record the match if any prefix of the word matches
                if j > 0:
                    start_pos = i - j + 1
                    if start_pos < N:
                        matches[start_pos].append(j)
                # If the entire word is matched, reset j
                if j == len_word:
                    j = prefix[j - 1]

        # Initialize DP array
        dp = [float('inf')] * (N + 1)
        dp[0] = 0  # Base case: no prefixes needed to form an empty string

        # Populate DP array
        for i in range(N):
            if dp[i] == float('inf'):
                continue  # Skip unreachable positions
            for l in matches[i]:
                if i + l > N:
                    continue  # Prevent index out of bounds
                if dp[i] + 1 < dp[i + l]:
                    dp[i + l] = dp[i] + 1

        # Return the result
        return dp[N] if dp[N] != float('inf') else -1

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