<p>Given the strings <code>s1</code> and <code>s2</code> of size <code>n</code> and the string <code>evil</code>, return <em>the number of <strong>good</strong> strings</em>.</p>

<p>A <strong>good</strong> string has size <code>n</code>, it is alphabetically greater than or equal to <code>s1</code>, it is alphabetically smaller than or equal to <code>s2</code>, and it does not contain the string <code>evil</code> as a substring. Since the answer can be a huge number, return this <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

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

<pre>
<strong>Input:</strong> n = 2, s1 = &quot;aa&quot;, s2 = &quot;da&quot;, evil = &quot;b&quot;
<strong>Output:</strong> 51 
<strong>Explanation:</strong> There are 25 good strings starting with &#39;a&#39;: &quot;aa&quot;,&quot;ac&quot;,&quot;ad&quot;,...,&quot;az&quot;. Then there are 25 good strings starting with &#39;c&#39;: &quot;ca&quot;,&quot;cc&quot;,&quot;cd&quot;,...,&quot;cz&quot; and finally there is one good string starting with &#39;d&#39;: &quot;da&quot;.&nbsp;
</pre>

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

<pre>
<strong>Input:</strong> n = 8, s1 = &quot;leetcode&quot;, s2 = &quot;leetgoes&quot;, evil = &quot;leet&quot;
<strong>Output:</strong> 0 
<strong>Explanation:</strong> All strings greater than or equal to s1 and smaller than or equal to s2 start with the prefix &quot;leet&quot;, therefore, there is not any good string.
</pre>

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

<pre>
<strong>Input:</strong> n = 2, s1 = &quot;gx&quot;, s2 = &quot;gz&quot;, evil = &quot;x&quot;
<strong>Output:</strong> 2
</pre>

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

<ul>
	<li><code>s1.length == n</code></li>
	<li><code>s2.length == n</code></li>
	<li><code>s1 &lt;= s2</code></li>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>1 &lt;= evil.length &lt;= 50</code></li>
	<li>All strings consist of lowercase English letters.</li>
</ul>

from functools import lru_cache

MOD = 1000000007

class Solution:
    def findGoodStrings(self, n: int, s1: str, s2: str, evil: str) -> int:
        s1 = list(map(ord, s1))
        s2 = list(map(ord, s2))
        evil = list(map(ord, evil))
        m = len(evil)
        pi = [0] * m
        p = [0] * (n + 1)

        # Compute the prefix function for KMP
        length = 0
        i = 1
        while i < m:
            if evil[i] == evil[length]:
                length += 1
                pi[i] = length
                i += 1
            elif length > 0:
                length = pi[length - 1]
            else:
                pi[i] = 0
                i += 1

        # Compute powers of 26 modulo MOD
        p[0] = 1
        for i in range(1, n + 1):
            p[i] = (26 * p[i - 1]) % MOD

        @lru_cache(None)
        def dp(i: int, j: int, b1: int, b2: int) -> int:
            if j == m:
                return 0
            if i == n:
                return 1

            lo = s1[i] - ord('a') if b1 > 0 else 0
            hi = s2[i] - ord('a') if b2 > 0 else 25
            res = 0

            for c in range(lo, hi + 1):
                ni = i + 1
                nj = j
                while nj > 0 and evil[nj] != c + ord('a'):
                    nj = pi[nj - 1]
                if c == evil[nj] - ord('a'):
                    nj += 1
                res = (res + dp(ni, nj, b1 & (c == lo), b2 & (c == hi))) % MOD

            return res

        return dp(0, 0, 1, 1)

# Example usage:
solution = Solution()
print(solution.findGoodStrings(3, "abc", "def", "cd"))  # Example usage

21700