
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 can be divided into two disjoint subsets $\{f(1), f(2), \cdots, f(n), \cdots\}$ and $\{g(1), g(2), \cdots, g(n), \cdots\}$, where

\[
f(1)<f(2)<\cdots<f(n)<\cdots
\]
\[
g(1)<g(2)<\cdots<g(n)<\cdots
\]

and $\quad g(n) = f(f(n)) + 1 \quad (n \geq 1)$

Find $f(240)$.

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