
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 $\{a\_n\}$ whose first $n$ terms sum $S\_n$ satisfies: $S\_1=1$, $3S\_n=(n+2)a\_n$.
(1) Find the values of $a\_2$ and $a\_3$;
(2) Find the general term formula of the sequence $\{a\_n\}$;
(3) Find the sum of $\frac{1}{a\_1} + \frac{1}{a\_2} + \ldots + \frac{1}{a\_n}$.

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