Let \( f \) be a function from the positive integers to the positive integers such that:
(i) \( f \) is strictly increasing (i.e., \( f(n+1) > f(n) \) for all positive integers \( n \)),
(ii) \( f(f(n)) = n \) for all positive integers \( n \),
(iii) \( f(n) \leq 2n \) for all positive integers \( n \).

Determine the smallest positive integer \( k \) such that \( f(k) = k \). Let's think step by step and output the final answer within \boxed{}.
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