
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 that the first term of an arithmetic sequence $\{a\_n\}$ is $1$ and the common difference is not $0$, the 2nd, 4th, and 9th terms form a geometric sequence. Find the common ratio $q$ of this geometric sequence, the general term formula $a\_n$ of the arithmetic sequence, and the sum of the first $n$ terms $S\_n$.

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