RO-NMO 2019 Problem 93

Prove that the number of solutions in $\left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)$ of the parametric equation $$\sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z,$$ is greater than zero and finite, for nay natural number $n.$