
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
Consider an arithmetic sequence $\{a_n\}$ with a common difference $d$ (not equal to zero) such that the sum of its first $n$ terms is $S_n$. Given that $S_3 = 9$, and that the terms $a_3$, $a_5$, and $a_8$ form a geometric sequence, find the common difference $d$, and express the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n$, in terms of $n$.

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