
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
Points \( A, B, C, \) and \( D \) are positioned on a line in the given order. It is known that \( BC = 3 \) and \( AB = 2 \cdot CD \). A circle is drawn through points \( A \) and \( C \), and another circle is drawn through points \( B \) and \( D \). Their common chord intersects segment \( BC \) at point \( K \). Find \( BK \).

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