
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 that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$ ($S_{n}\neq 0$), and $T_{n}$ is the product of the first $n$ terms of the sequence $\{S_{n}\}$, satisfying $S_{n}+T_{n}=S_{n}\cdot T_{n}$ for $n\in \mathbb{N}^{*}$, the following four statements are given: <br/>①$a_{1}=2$; ②$a_{n}=\frac{2}{n(2n-1)}$; ③$\{T_{n}\}$ is an arithmetic sequence; ④$S_{n}=\frac{n+1}{n}$. The correct statements are those with the following numbers: ____.

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