
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 $f(1)<f(2)<\cdots<f(n)<\cdots$ and $g(1)<g(2)<\cdots<g(n)<\cdots$ satisfy the condition $g(n)=f[f(n)]+1$ for $n \geq 1$. Find $f(240)$.

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