
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 $n \in \mathbb{N}^{*}$, the sequence $\{a_n\}$ has its first term $a_1=1$, and the function $f(x) = \frac{1}{3}x^{3}-(a_{n}+n+3)x^{2}+2(2n+6)a_{n}x$. If $x=a_{n+1}$ is a point of local minimum for $f(x)$, determine the general formula for the sequence $\{a_n\}$.

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