
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
Let \( A, B, C, \) and \( D \) be four points lying in this order on a circle. Assume that there is a point \( K \) on the line \( AB \) such that \( BD \) bisects the line \( KC \) and \( AC \) bisects the line \( KD \). Determine the smallest possible value that \(\left|\frac{AB}{CD}\right|\) can take.

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