CMO 2022 Problem 1

Assume that real numbers $a$ and $b$ satisfy

$$a b+\sqrt{a b+1}+\sqrt{a^{2}+b} \cdot \sqrt{b^{2}+a}=0$$

Find, with proof, the value of

$$a \sqrt{b^{2}+a}+b \sqrt{a^{2}+b}$$