
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 $\{a_n\}$ be a positive geometric sequence, and let $S_n = \log a_1 + \log a_2 + \ldots + \log a_n$, where $n \in \mathbb{N}^*$. If there exist distinct positive integers $m$ and $n$ such that $S_m = S_n$, then $S_{m+n} = \_\_\_\_\_\_$.

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