
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
The chord \( AB \) of a sphere with a radius of 1 has a length of 1 and is located at an angle of \( 60^\circ \) to the diameter \( CD \) of this sphere. It is known that \( AC = \sqrt{2} \) and \( AC < BC \). Find the length of the segment \( BD \).

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