
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 sets $A=\{x\,|\,|2x-1|\leq3\}$ and $B=\{x\,|\,|x+2|<1\}$, is there a set $C$ that simultaneously satisfies the following three conditions:
1. $C$ contains 3 elements;
2. $C \cap B \neq \emptyset$;
3. $C \subseteq [(A \cup B) \cap \mathbb{Z}]$.

If such a set exists, find the set $C$; if it does not exist, explain why.

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