Integrals I: Feynman's Trick · Superchallenge problem

Positive integers $a, b$ satisfy $\dfrac{a^2 + b^2}{ab + 1} = k$ for an integer $k$. Vieta jumping shows $k$ must be a perfect square. Find the smallest solution $(a, b)$ with $a > b > 0$ and $k = 4$, and give $a + b$.