SMO 2022 Open Round 1 Question 25

Let $\mathbb R^+$ be the set of all positive real numbers. Let $f:\mathbb R^+\to\mathbb R^+$ satisfy $$xyf(x)\bigl(f(y)-f(yf(x))\bigr)=1$$ for all $x,y\in\mathbb R^+$. Find $f(1/2022)$.