
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
Suppose that $\{a\_n\}$ is a positive geometric sequence, and let $S\_n = \lg a\_1 + \lg a\_2 + ... + \lg a\_n$, where $n \in \mathbb{N}^*$. If there exist distinct positive integers $m$ and $n$ such that $S\_m = S\_n$, then $S\_{m+n} =$ $\boxed{\text{answer}}$.

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