
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 line passing through the common point $A$ of two circles intersects these circles again at points $B$ and $C$, respectively. The distance between the projections of the centers of the circles onto this line is 12. Find $BC$, given that point $A$ lies on the segment $BC$.

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