CGMO 2009 Problem 3

Let $n$ be a given positive integer. In the coordinate set, consider the set of points $\{P_{1},P_{2},...,P_{4n+1}\}=\{(x,y)|x,y\in \mathbb{Z}, xy=0, |x|\le n, |y|\le n\}.$

Determine the minimum of $(P_{1}P_{2})^{2} + (P_{2}P_{3})^{2} +...+ (P_{4n}P_{4n+1})^{2} + (P_{4n+1}P_{1})^{2}.$