
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
Let \( f, g: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) be strictly increasing functions such that 
\[ f(\mathbf{N}^{*}) \cup g(\mathbf{N}^{*}) = \mathbf{N}^{*} \] 
and 
\[ f(\mathbf{N}^{*}) \cap g(\mathbf{N}^{*}) = \varnothing \] 
with 
\[ g(n) = f(f(n)) + 1. \]

Find \( f(240) \).

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