OIMU 2005 Problem 3

Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$ , $(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$ .

Find $\lim_{n\to \infty}(x_n,y_n)$ .