9.8 Dima calculated the reciprocal of the factorial of each integer from 80 to 100. He printed the resulting decimal fractions on 19 infinitely long strips of paper (for example,

Sasha cut a segment from one of the strips, which had exactly $n$ digits without a decimal point. If Sasha doesn't want Dima to guess which strip the $n$ digits were cut from, what is the maximum value of $n$?
The maximum value of $n$ is 155.
Assuming that on slips of paper written with $\frac{1}{k!}$ and $\frac{1}{l!}$ (where $k < l$), there is a segment of 156 digits that is the same, then the difference between these two numbers is less than $\frac{1}{10^{156}}$. Therefore, $n < 156$.
In this way, for any segment of 156 digits, we can determine its origin. On the other hand, on slips of paper written with $\frac{1}{98!}$ and $\frac{1}{99!}$, there is a completely identical segment of 155 digits: