
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 the sequence \(\{a_n\}\) defined by \(a_{1}=\frac{3}{2}\), and \(a_{n}=\frac{3 n a_{n-1}}{2 a_{n-1}+n-1}\) for \(n \geq 2\), \(n \in \mathbb{N}_{+}\).

(1) Find the general term of the sequence \(\{a_n\}\).

(2) For all positive integers \(n\), the inequality \(a_{1} a_{2} a_{3} \cdots a_{n}<\lambda \cdot n!\) always holds. Determine the smallest positive integer \(\lambda\).

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