
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 \( f: \mathbb{Z}^* \rightarrow \mathbb{R} \) that satisfy the equation
\[ f(n+m) + f(n-m) \equiv f(3n) \]
for \( n, m \in \mathbb{Z}^* \) (where \(\mathbb{Z}^*\) is the set of non-negative integers) and \( n \geq m \).

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