Inspired by Aakash Khandelwal

Geometry Level 5

If $a_0=\sin^2\left (\dfrac{\pi}{45} \right)$ and $a_{n+1}=4a_{n}(1-a_{n})$ for $n\geq 0$ , find the smallest positive integer $n$ such that $a_n=a_0$ .

If you come to the conclusion that no such $n$ exists, enter 666.