BALTIC-WAY 2022 Problem 4

The positive real numbers $x,y,z$ satisfy $xy+yz+zx=1$ . Prove that: $$2(x^2+y^2+z^2)+\frac{4}{3}\bigg (\frac{1}{x^2+1}+\frac{1}{y^2+1}+\frac{1}{z^2+1}\bigg) \ge 5$$