Ball in a hexagon

In the Cartesian coordinate system $OXY$, a regular hexagon is drawn. Each side of the hexagon has length $1$ and each angle is $120$ degrees (because it is regular). The center of the hexagon is at $(0,0)$, and two of its vertices are at $(1,0)$ and $(-1,0)$.

A small round ball is initially placed at $(0,0)$ and will be treated as a point mass. The ball is struck so that it starts moving at a constant speed of one unit segment per second (where a unit segment is the segment connecting $(0,0)$ and $(1,0)$) in the direction making a $45$-degree angle with the positive $OX$ axis, as shown below:

If, during its motion, the ball touches any side of the hexagon, it is reflected from that side according to the law of reflection—the angle of incidence equals the angle of reflection. The collision is perfectly elastic, so after the reflection, the ball continues to move at the same speed.

If the ball hits a corner of the hexagon exactly, the reflection also occurs according to physical laws: the angle of incidence is measured from one side forming the corner and the angle of reflection from the other.

Assume that after exactly $T$ seconds of motion, the ball is at the point (x_0, y_0). Write a program that takes $T$ as input and finds the value of x_0^2 + y_0^2.