
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
Two circles intersect at points \( A \) and \( B \). Diameters \( AC \) and \( AD \) are drawn through point \( A \) in these circles. Find the absolute value of the difference between segments \( BC \) and \( BD \), if the distance between the centers of the circles is \( a \), and the centers of the circles lie on the same side of the common chord \( AB \).

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