
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 a sequence $\left\{a_{n}\right\}$ with the partial sum $S_{n}$ satisfying the condition
\[ S_{n} + S_{n+1} + S_{n+2} = 6n^2 + 9n + 7 \quad (n \geq 1), \]
and it is known that $a_{1} = 1$, $a_{2} = 5$, find the general term formula for the sequence $\left\{a_{n}\right\}$.

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