
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
Given an arithmetic-geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. It is known that $S\_1=16$. A student calculated and obtained $S\_2=32$, $S\_3=76$, $S\_4=130$. Upon verification, it was found that exactly one of these numbers was incorrect. Determine the incorrect number and the common ratio of the sequence.

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