
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
Convex quadrilateral \( BCDE \) lies in the plane. Lines \( EB \) and \( DC \) intersect at \( A \), with \( AB = 2 \), \( AC = 5 \), \( AD = 200 \), \( AE = 500 \), and \(\cos \angle BAC = \frac{7}{9} \). What is the largest number of nonoverlapping circles that can lie in quadrilateral \( BCDE \) such that all of them are tangent to both lines \( BE \) and \( CD \)?

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