International Mathematical Olympiad 1978 Problem 3

The set of all positive integers is the union of two disjoint subsets $\{f(1), f(2), \ldots, f(n), \ldots\}$, $\{g(1), g(2), \ldots, g(n), \ldots\}$, where

$$f(1) < f(2) < \cdots < f(n) < \cdots,$$ $$g(1) < g(2) < \cdots < g(n) < \cdots,$$

and

$$g(n) = f(f(n)) + 1 \text{ for all } n \geq 1.$$

Determine $f(240)$.