Lengths and Periods

In mathematics and computer science, the critical exponent of a string is the largest number of times a contiguous substring repeats in a row. The value can be a fraction. For example, the critical exponent of "Mississippi" is $7/3$, because it contains the substring "ississi", which has length 7 and period 3. Let $w$ and $x$ be non-empty strings. $x$ occurs in $w$ with exponent $\alpha$, for a positive rational $\alpha$, if $w$ has a substring $y$ with $y = x^n x_0$, where $x^n$ is $x$ repeated $n$ times, $x_0$ is a prefix of $x$, $n$ is the integer part of $\alpha$, and $|y| = \alpha |x|$. The critical exponent of $w$ is the maximum $\alpha$ over all $x^\alpha$ that occur in $w$.

The only line contains a string $w$, a sequence of lowercase English letters, where $1 \le |w| \le 200\,000$.

Output the critical exponent of $w$ as an irreducible fraction $p/q$, where $p$ and $q$ are integers without leading zeros.