CMO 2015 Problem 3

On a $(4 n+2) \times(4 n+2)$ square grid, a turtle can move between squares sharing a side. The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?