Let $f(x, y) = \frac{x + y}{2}$, $g(x, y) = \sqrt{xy}$, $h(x, y) = \frac{2xy}{x + y}$, and let
$$ S = \{ (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \neq b \text{ and } f(a, b), g(a, b), h(a, b) \in \mathbb{N} \} $$
where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$.

Let's think step by step and output the final answer within \boxed{}.
user