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 ...