Yet another curious identity?

Geometry Level 5

Find the smallest integer \(n>3\) such that \(\displaystyle \prod_{k=3}^{n}2\cos(2^k)=1\), where angles are measured in degrees.

If no such \(n\) exists, enter 666 as your answer.

Bonus question: What is the solution for \(\displaystyle \prod_{k=2}^{n}2\cos(2^k)=1\)?

Inspiration

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