
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 the sequence $\{a_n\}$, solve for $a_n$ given the following conditions:
1. If $a_1 = 1$ and $a_n = 3a_{n-1} + 1$, find $a_n$.
2. If $S_n = 2n^2 - 3n + 1$, find $a_n$ where $S_n$ denotes the sum of the first $n$ terms of the sequence.

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