Brilliant Math Contest

Algebra Level 5

To celebrate the new year, the Brilliant society decided to host a math contest. The results were fantastic, and Brilliant decided to award the top \(n\) students with a medal \((n>1)\). For the first round, Brilliant awarded it to the top student, along with \({1}/{7}\) of the remaining students (i.e. \(n-1\) of them).
For the second round, Brilliant awarded it to the remaining top two students, along with \(1/7\) of the remaining students.
This process continued, until the \(k^{th}\) award, which was awarded to the remaining top \(k\) students. However, at this point in time, there were no more medals left.

How many students won a medal? (a.k.a. what is the value of \(n\)?)

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