
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
Suppose $S$ is a non-empty subset of the real number set $\mathbb{R}$. If for any $x, y \in S$, we have $x+y$, $x-y$, $xy \in S$, then $S$ is called a closed set. Consider the following statements:
1. The set $S=\{a+b\sqrt{3}\mid a,b \text{ are integers}\}$ is a closed set.
2. If $S$ is a closed set, then it must contain 0.
3. A closed set must have infinitely many elements.
4. If $S$ is a closed set and $S \subseteq T \subseteq \mathbb{R}$ for any set $T$, then $T$ is also a closed set.

Select all correct statements.

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