Problem: Find the Winner of the Circular Game

Statement

A group of n friends is seated in a circle, numbered from $1$ to n in clockwise order. Specifically, moving clockwise from friend i leads to friend i + 1 for $1 \leq$ i $<$ n, and moving clockwise from friend n wraps back to friend $1$.

The game proceeds as follows:

1. Begin at friend $1$.

2. Starting from the current friend, count k friends in the clockwise direction (including the current friend). The counting wraps around the circle and may count some friends more than once.

3. The last friend counted is eliminated ...