<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>

class Solution:
    def helper(self, mid, x, n):
        lhs = mid ** x
        rhs = n * mid - n + 1
        return lhs <= rhs

    def helperr(self, mid, x, n):
        lhs = mid ** x
        rhs = n * mid - n + 1
        return lhs == rhs

    def smallestGoodBase(self, n: str) -> str:
        num = int(n)
        ans = num - 1
        for x in range(2, 61):
            l = 2
            r = num - 1
            c = num - 1
            while l <= r:
                mid = (l + r) // 2
                if self.helper(mid, x, num):
                    l = mid + 1
                    if self.helperr(mid, x, num):
                        c = mid
                else:
                    r = mid - 1
            ans = min(ans, c)
        return str(ans)

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