There are 3 questions in a test and the full mark of each question is 7. Each question can only be marked with integers: $1, 2, 3, \cdots, 7$. We know that the product of everyone's marks of the 3 questions is 36 (i.e., the product of the sums of the marks of the three questions across everyone is 36) and that the marks of any two people are not exactly the same. How many people did the test at most?
What I have done:
Assume there are $n$ people, let $x_i$ be the sum of the three questions for person $i$ where $i = 1, 2, \cdots, n$. We know that $x_1x_2x_3 \cdots x_n=36$ and that $x_i \neq x_j$ for $i \neq j$. How can we find the largest possible $n$?