RS-SMO 2012 Problem 2

Find all natural numbers $a$ and $b$ such that $a|b^2, b|a^2 \text{ and } a+1|b^2+1.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.