
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
A segment \( AB \) (with \( AB = 1 \)), which is a chord of a sphere with radius 1, is positioned at an angle of \( 60^\circ \) to the diameter \( CD \) of the sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Find \( BD \).

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