
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 $\mathbf{N}^{*} = \{1, 2, 3, \cdots\}$. Determine whether there exists a function $f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ such that:
(1) $f(1) = 2$;
(2) $f(f(n)) = f(n) + n$ holds for all $n \in \mathbf{N}^{*}$;
(3) $f(n) < f(n+1)$ holds for all $n \in \mathbf{N}^{*}$.

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