Symmetric polynomials; $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}\implies a_1=\dots =a_n=0$

Question

Let $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_\limits{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}$.

Prove that $a_1=a_2=\dots=a_n=0$.

I have tried to show that all symmetric elementary polynomials determined by $a_{1},a_{2}\dots a_{n}$ are equal to $0$. My guess is that I have to expand, and use Newton sums to express the L.H.S in terms of the symmetric elementary polynomials, but the equations don't look very nice.