ELMO SL 2018 A3

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that $$a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.$$

Proposed by Daniel Liu