
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 \( n \geq 2 \), find the minimum value of \( |X| \) such that for any \( n \) binary subsets \( B_1, B_2, \ldots, B_n \) of set \( X \), there exists a subset \( Y \) of \( X \) satisfying:
1. \( |Y| = n \);
2. For \( i = 1, 2, \ldots, n \), \( |Y \cap B_i| \leq 1 \).

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