HMMT 2019 T10

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial

$$P(x)=(2 n) x^{2 n}+(2 n-1) x^{2 n-1}+\cdots+(n+1) x^{n+1}+n x^{n}+(n+1) x^{n-1}+\cdots+(2 n-1) x+2 n$$

lie on the unit circle (i.e. $|r|=1$ ).