<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>

class Solution:
    def minimumBeautifulSubstrings(self, s: str) -> int:
        # Generate all powers of 5 up to 5^15 (since s.length <= 15)
        powers_of_five = {pow(5, i) for i in range(16)}
        
        # Function to check if a substring is beautiful
        def is_beautiful(substring):
            return substring[0] != '0' and int(substring, 2) in powers_of_five
        
        # Backtracking function to find the minimum number of beautiful substrings
        def backtrack(start):
            if start == len(s):
                return 0
            
            min_partitions = float('inf')
            for end in range(start + 1, len(s) + 1):
                substring = s[start:end]
                if is_beautiful(substring):
                    partitions = backtrack(end)
                    if partitions != -1:
                        min_partitions = min(min_partitions, 1 + partitions)
            
            return min_partitions if min_partitions != float('inf') else -1
        
        return backtrack(0)

# Example usage:
solution = Solution()
print(solution.minimumBeautifulSubstrings("1011"))  # Output: 2
print(solution.minimumBeautifulSubstrings("111"))   # Output: 3
print(solution.minimumBeautifulSubstrings("0"))     # Output: -1
60