
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
Find all functions \( f: \mathbf{Z}^{*} \rightarrow \mathbf{R} \) (where \(\mathbf{Z}^{*}\) is the set of non-negative integers) that satisfy \( f(n+m) + f(n-m) = f(3n) \) for all \( m, n \in \mathbf{Z}^{*} \) with \( n \geqslant m \).

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