
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 $n (\geqslant 2)$, find the minimum value of $|X|$ such that for any $n$ two-element subsets $B_{1}, B_{2}, \cdots, B_{n}$ of the set $X$, there exists a subset $Y$ of $X$ satisfying:
(1) $|Y| = n$;
(2) For $i = 1, 2, \cdots, n$, we have $\left|Y \cap B_{i}\right| \leqslant 1$.
Here, $|A|$ represents the number of elements in the finite set $A$.

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