
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
In a trapezoid \(ABCD\) with bases \(AD=12\) and \(BC=8\), circles constructed on the sides \(AB\), \(BC\), and \(CD\) as diameters intersect at one point. The length of diagonal \(AC\) is 12. Find the length of \(BD\).

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