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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.

Inspiration.

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