
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 a geometric series $\{a_n\}$ with the sum of the first $n$ terms as $S_n$, and if $S_2 + a_2$, $S_1 + 2a_2$, $S_3 + a_3$ form an arithmetic sequence, determine the common ratio of the series $\{a_n\}$.

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