
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
If \( A_{1}, A_{2}, \cdots, A_{k} \) is a cover of \( X \), and each element of \( X \) belongs to exactly two of the sets \( A_{1}, A_{2}, \cdots, A_{k} \), then \( A_{1}, A_{2}, \cdots, A_{k} \) is called a double cover of \( X \). Determine the number of double covers for \( k=3 \).

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