<p>Given a binary string <code>s</code>, partition the string into one or more <strong>substrings</strong> such that each substring is <strong>beautiful</strong>.</p>

<p>A string is <strong>beautiful</strong> if:</p>

<ul>
	<li>It doesn&#39;t contain leading zeros.</li>
	<li>It&#39;s the <strong>binary</strong> representation of a number that is a power of <code>5</code>.</li>
</ul>

<p>Return <em>the <strong>minimum</strong> number of substrings in such partition. </em>If it is impossible to partition the string <code>s</code> into beautiful substrings,&nbsp;return <code>-1</code>.</p>

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

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

<pre>
<strong>Input:</strong> s = &quot;1011&quot;
<strong>Output:</strong> 2
<strong>Explanation:</strong> We can paritition the given string into [&quot;101&quot;, &quot;1&quot;].
- The string &quot;101&quot; does not contain leading zeros and is the binary representation of integer 5<sup>1</sup> = 5.
- The string &quot;1&quot; does not contain leading zeros and is the binary representation of integer 5<sup>0</sup> = 1.
It can be shown that 2 is the minimum number of beautiful substrings that s can be partitioned into.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;111&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> We can paritition the given string into [&quot;1&quot;, &quot;1&quot;, &quot;1&quot;].
- The string &quot;1&quot; does not contain leading zeros and is the binary representation of integer 5<sup>0</sup> = 1.
It can be shown that 3 is the minimum number of beautiful substrings that s can be partitioned into.
</pre>

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

<pre>
<strong>Input:</strong> s = &quot;0&quot;
<strong>Output:</strong> -1
<strong>Explanation:</strong> We can not partition the given string into beautiful substrings.
</pre>

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

<ul>
	<li><code>1 &lt;= s.length &lt;= 15</code></li>
	<li><code>s[i]</code> is either <code>&#39;0&#39;</code> or <code>&#39;1&#39;</code>.</li>
</ul>

from math import log, inf
from functools import cache

verbose = True

class Solution:
    def minimumBeautifulSubstrings(self, s: str) -> int:
        # for i in range(len(s)-1):
        #     for j in range(i, len(s)):
        #         tmp = s[i:j+1]
        #         print(f"{tmp} -> {int(tmp, 2)}")

        # return -1

        assert self.is_beautiful('1') == True
        assert self.is_beautiful('101') == True

        ret = self.helper(s)
        if ret == inf:
            return -1
        return ret

    @cache
    def helper(self, s):
        if len(s) == 0 or s[0] == '0': return inf
        if self.is_beautiful(s):
            return 1
        
        min_count = inf
        for i in range(1, len(s)):
            a_s = s[:i]
            b_s = s[i:]
            a = self.helper(a_s)
            b = self.helper(b_s)
            if verbose: print(f"{i=} {s=} {a_s=} {a=} {b_s=} {b=}")
            if max(a, b) == inf: continue
            min_count = min(min_count, a+b)
        return min_count

    
    @staticmethod
    # @cache
    def is_beautiful(s):
        ret = (s[0] != '0') and (log(int(s, 2), 5) % 1 < 0.000000001)
        # if verbose: print(f"{s}: {ret}")
        return ret
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