PAMO 2022 Problem 4

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$(f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2$$ $$(-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1)$$ Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.