linear algebra - Proving ($left|left|Axright|right| = left|left|xright|right|$, for all $xinmathbb{C}^n$) $implies A$ is unitary

As the title states, I'm trying to prove that $\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$ for all $x\in\mathbb{C}^n\implies$ $A$ is unitary, where $A$ is a square matrix.

This is part of a larger problem and the only thing I 'know' for the purposes of this proof is that $\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$ for all $x\in\mathbb{C}^n$, nothing else, e.g. I don't know that the rows or columns of $A$ are orthonormal.

Any hints on where I can start with this? I'm quite stuck.

Thank you very much for your time.