Let $W$ be a fixed positive integer. Let $S$ be the set of all pairs $(a, b)$ of positive integers such that $a \neq b$. For each $(a, b) \in S$, let $m(a,b)$ be the largest integer satisfying \[ m(a, b) \leq \frac{1 + na}{1 + nb} \]for all integers $n \geq 1$. (a) For each $(a, b) \in S$, prove that there exists a positive integer $k$ such that \[ m(a,b) \leq \frac{1 + na}{W + nb} \]for all $n \geq k$. (b) For each $(a, b) \in S$, let $k(a,b)$ be the smallest value of $k$ that satisfies the condition of part (a). Determine $\max \{k(a,b) \mid (a,b) \in S \}$ or prove that it does not exist.