<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:
        n, primes, mod = len(s), set('2357'), (10 ** 9) + 7
        @cache
        def dp(i, at_start, k):
            if i == n: return int(k == 0)
            if i > n or k == 0 or s[i] not in primes and at_start: return 0
            if s[i] in primes:
                if at_start: return dp(i + minLength - 1, False, k)
                else: return dp(i + 1, False, k)
            return (dp(i + 1, True, k - 1) + dp(i + 1, False, k)) % mod
        return dp(0, True, k)
        
545100