There are 8 lily pads in a pond numbered 1,2,…,8. A frog starts on lily pad 1. During the $i$-th second, the frog jumps from lily pad $i$ to $i+1$, falling into the water with probability $\frac{1}{i+1}$. The probability that the frog lands safely on lily pad 8 without having fallen into the water at any point can be written as $\frac{m}{n}$, where $m,n$ are positive, coprime integers. Find $100m+n$.

Source: Harvard-MIT Math Tournament, Guts Round, November 2021

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