BALTIC-WAY 2024 Problem 20

Positive integers $a, b$ and $c$ satisfy the system of equations

$$\left\{\begin{aligned} (a b-1)^{2} & =c\left(a^{2}+b^{2}\right)+a b+1 \\ a^{2}+b^{2} & =c^{2}+a b \end{aligned}\right.$$

(a) Prove that $c+1$ is a perfect square. (b) Find all such triples $(a, b, c)$.