Interactive Factorial Guessing

Oh no, this wicked jury hides something from you again, and you need to guess it interactively.

This time, you need to find an integer $n$. To do that, you can make at most 10 queries of the form "What is the $k$-th decimal digit of the product of all integers from 1 to $n$ (also known as factorial and denoted as $n!$)?".