
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
For the sequence $\{a_n\}$, the sum of its first $n$ terms is $S_n = n^2 + \frac{1}{2}n$. Find the general term formula $a_n$.

For another sequence $\{a_n\}$ with the sum of its first $n$ terms being $S_n = \frac{1}{4}n^{2} + \frac{2}{3}n + 3$, find the general term formula $a_n$.

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