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.