<p>Given an integer <code>n</code> represented as a string, return <em>the smallest <strong>good base</strong> of</em> <code>n</code>.</p>

<p>We call <code>k &gt;= 2</code> a <strong>good base</strong> of <code>n</code>, if all digits of <code>n</code> base <code>k</code> are <code>1</code>&#39;s.</p>

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

<pre>
<strong>Input:</strong> n = &quot;13&quot;
<strong>Output:</strong> &quot;3&quot;
<strong>Explanation:</strong> 13 base 3 is 111.
</pre>

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

<pre>
<strong>Input:</strong> n = &quot;4681&quot;
<strong>Output:</strong> &quot;8&quot;
<strong>Explanation:</strong> 4681 base 8 is 11111.
</pre>

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

<pre>
<strong>Input:</strong> n = &quot;1000000000000000000&quot;
<strong>Output:</strong> &quot;999999999999999999&quot;
<strong>Explanation:</strong> 1000000000000000000 base 999999999999999999 is 11.
</pre>

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

<ul>
	<li><code>n</code> is an integer in the range <code>[3, 10<sup>18</sup>]</code>.</li>
	<li><code>n</code> does not contain any leading zeros.</li>
</ul>

def powsum(base: int, digit: int) -> int:
    s = 0
    for _ in range(digit):
        s *= base
        s += 1
    return s


class Solution:

    def find_valid_base(self, digit: int) -> Optional[int]:
        left = 2
        right = self.n

        while left <= right:
            base = (left + right) >> 1
            ps = powsum(base, digit)
            if ps == self.n:
                return base
            elif ps < self.n:
                left = base + 1
            else:
                right = base - 1

    def smallestGoodBase(self, n: str) -> str:
        self.n = int(n)
        n = self.n
        msb = 0
        while n:
            n >>= 1
            msb += 1
        
        for digit in reversed(range(1, msb + 1)):
            base = self.find_valid_base(digit)
            if base:
                return str(base)

        raise ValueError("Cannot find base")

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