Let $n=2^k$ where $k\ge 3$. Let $a$ be any odd natural number.

Prove that $a^{n/4}\equiv 1\pmod{n}$.

**My attempt**:

$\phi(n)=n/2$

So, by Euler's formula, $a^{n/2}\equiv 1\pmod{n}$.

I don't know how to proceed.