On the coordinate plane is given the square with vertices $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$. For every $n\in\mathbb N$, point $T_{n+4}$ is defined as the midpoint of the segment $T_nT_{n+1}$. Determine the coordinates of the limit point of $T_n$ as $n\to\infty$, if it exists.