
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 sequence $\{a_n\}$ with a nonzero common difference $d$, let $S_n = a_1 + a_2q + \dots + a_nq^{n-1}$, and $T_n = a_1 - a_2q + \dots + (-1)^{n-1}a_nq^{n-1}$, where $q \neq 0$ and $n \in \mathbb{N}^*$.
1. If $q=1$, $a_1=1$, and $S_3=15$, find the general term formula for the sequence $\{a_n\}$.
2. If $a_1=d$, and $S_1$, $S_2$, $S_3$ form a geometric sequence, find the value of $q$.

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