<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>

class Solution {
public:
    int findGoodStrings(int n, string s1, string s2, string evil) {
        int M = 1e9+7;
        vector<int> pi(evil.size());
        int match = 0;
        for (int i = 1; i < evil.size(); ++i) {
            while (match && evil[i] != evil[match]) {
                match = pi[match - 1];
            }
            if (evil[match] == evil[i]) {
                match++;
            }
            pi[i] = match;
        }

        int memo[n][evil.size() + 1];
        memset(memo, -1, sizeof(memo));
        function<int(int, int, bool, bool)> dfs = [&] (int i, int j, bool limit_lo, bool limit_high) {
            if (i == n) {
                if (j != evil.size()) {
                    return 1;
                }
                return 0;
            }

            if (!limit_lo && !limit_high) {
                int res = memo[i][j];
                if (res != -1) {
                    return res;
                }
            }

            int lo = limit_lo ? s1[i] - 'a' : 0;
            int hi = limit_high ? s2[i] - 'a' : 25;

            int res = 0;
            for (int d = lo; d <= hi; ++d) {
                if (j == evil.size()) {
                    res += dfs(i + 1, j, limit_lo && d == lo, limit_high && d == hi);
                } else {
                    if (d == (evil[j] - 'a')) {
                        res += dfs(i + 1, j + 1, limit_lo && d == lo, limit_high && d == hi);
                    } else {
                        int j_cp = j;
                        while (j_cp && d != (evil[j_cp] - 'a')) {
                            j_cp = pi[j_cp - 1];
                        }
                        if (d ==  (evil[j_cp] - 'a')) {
                            j_cp++;
                        }
                        res += dfs(i + 1, j_cp, limit_lo && d == lo, limit_high && d == hi);
                    }
                }
                res %= M;
            }
            if (!limit_lo && !limit_high) {
                memo[i][j] = res;
            }
            return res;
        };
        return dfs(0, 0, true, true);
    }
};
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