<p>You are given a string <code>s</code> that consists of the digits <code>&#39;1&#39;</code> to <code>&#39;9&#39;</code> and two integers <code>k</code> and <code>minLength</code>.</p>

<p>A partition of <code>s</code> is called <strong>beautiful</strong> if:</p>

<ul>
	<li><code>s</code> is partitioned into <code>k</code> non-intersecting substrings.</li>
	<li>Each substring has a length of <strong>at least</strong> <code>minLength</code>.</li>
	<li>Each substring starts with a <strong>prime</strong> digit and ends with a <strong>non-prime</strong> digit. Prime digits are <code>&#39;2&#39;</code>, <code>&#39;3&#39;</code>, <code>&#39;5&#39;</code>, and <code>&#39;7&#39;</code>, and the rest of the digits are non-prime.</li>
</ul>

<p>Return<em> the number of <strong>beautiful</strong> partitions of </em><code>s</code>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>A <strong>substring</strong> is a contiguous sequence of characters within a string.</p>

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

<pre>
<strong>Input:</strong> s = &quot;23542185131&quot;, k = 3, minLength = 2
<strong>Output:</strong> 3
<strong>Explanation:</strong> There exists three ways to create a beautiful partition:
&quot;2354 | 218 | 5131&quot;
&quot;2354 | 21851 | 31&quot;
&quot;2354218 | 51 | 31&quot;
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;23542185131&quot;, k = 3, minLength = 3
<strong>Output:</strong> 1
<strong>Explanation:</strong> There exists one way to create a beautiful partition: &quot;2354 | 218 | 5131&quot;.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;3312958&quot;, k = 3, minLength = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> There exists one way to create a beautiful partition: &quot;331 | 29 | 58&quot;.
</pre>

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

<ul>
	<li><code>1 &lt;= k, minLength &lt;= s.length &lt;= 1000</code></li>
	<li><code>s</code> consists of the digits <code>&#39;1&#39;</code> to <code>&#39;9&#39;</code>.</li>
</ul>

class Solution:
    def beautifulPartitions(self, s: str, k: int, minLength: int) -> int:
        MOD = 7 + 10 ** 9
        prime_digits = {"2", "3", "5", "7"}
        @cache
        def solve(index: int, k_count: int):
            if index >= len(s):
                if k_count == k:
                    return 1
                return 0
            if k_count > k or s[index] not in prime_digits:
                return 0
            remaining_length = max(0, (k - k_count - 1) * minLength)
            count = 0
            for i in range(index + minLength - 1, len(s) - remaining_length):
                if s[i] not in prime_digits:
                    count += solve(i + 1, k_count + 1)
            return count % MOD
        return solve(0, 0)
98500