
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 $\left\{a_{n}\right\}$ be a geometric sequence of positive terms. Define $S_{n}=\lg a_{1}+\lg a_{2}+\cdots+\lg a_{n}$ for all $n \in \mathbf{N}^{*}$. If there exist distinct positive integers $m$ and $n$ such that $S_{n}=S_{m}$, determine $S_{m+n}$.

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