
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The set of all positive integers is the union of two disjoint subsets \(\{f(1), f(2), f(3), ...\}\) and \(\{g(1), g(2), g(3), ...\}\), where \(f(1) < f(2) < f(3) < ...\), and \(g(1) < g(2) < g(3) < ...\), and \(g(n) = f(f(n)) + 1\) for \(n = 1, 2, 3, ... \). Determine \(f(240)\).

Present the answer in LaTex format: \boxed{Your answer}
user