
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 $\{a_{n}\}$ where $a_{1}=4$ and $na_{n+1}=2(n+1)a_{n}$, the general formula for the sequence $\{a_{n}\}$ is ____, and if the sum of the first $n$ terms of the sequence $\left\{\frac{a_n}{(n+1)(n+2)}\right\}$ is $S_{n}$, then the smallest value of $n$ that satisfies the inequality $S_{n}\geqslant 30$ is ____.

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