
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 positive integer  $ n\geq 2$ , let  $ B_{1}$ ,  $ B_{2}$ , ...,  $ B_{n}$  denote  $ n$  subsets of a set  $ X$  such that each  $ B_{i}$  contains exactly two elements. Find the minimum value of  $ \left|X\right|$  such that for any such choice of subsets  $ B_{1}$ ,  $ B_{2}$ , ...,  $ B_{n}$ , there exists a subset  $ Y$  of  $ X$  such that:
 (1)  $ \left|Y\right| \equal{} n$ ;
 (2)  $ \left|Y \cap B_{i}\right|\leq 1$  for every  $ i\in\left\{1,2,...,n\right\}$ .

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