
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 $n \in \mathbb{N}^*$, and $f(n) = 5^n + 2 \times 3^{n-1} + 1$. By calculating the values of $f(n)$ for $n=1, 2, 3, 4$, we can guess the largest integer that $f(n)$ is divisible by.

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