
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
In the sequence $\{a_n\}$, $a_1=\frac{1}{2}$, and $(n+1)(na_n\cdot a_{n+1}+a_{n+1})-na_n=0$ $(n\in\mathbb{N}^*)$. Let the sum of the first $n$ terms of the sequence $\left\{\frac{a_n}{n+2}\right\}$ be $S_n$. Find $S_n$.

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