A regular hexagon of side length $a=6$ is present inside a larger regular hexagon of side length $b=18$. Initially, the smaller hexagon is placed such that one of its vertices coincides with that of the larger hexagon. Now, the smaller hexagon starts moving at an angle $\theta$ from one of the touching edges such that $\tan(\theta)=11\sqrt{3}$.

If any edge of the smaller hexagon touches the larger hexagon, then the smaller hexagon "reflects off" (see GIF for clarification) according to laws of reflection, and the smaller hexagon always stays inside the larger one.

However, if any vertex of the smaller hexagon coincides with any vertex of the larger hexagon, then the smaller hexagon stops moving. If the total distance traveled by the centre of the smaller hexagon until it stops moving is $d$, then give the value of $d^2$.

Note: Data is given such that it will stop moving.

Here is a gif demonstrating the motion of the smaller hexagon :-