
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
Let \( AEF \) be a triangle with \( EF = 20 \) and \( AE = AF = 21 \). Let \( B \) and \( D \) be points chosen on segments \( AE \) and \( AF \), respectively, such that \( BD \) is parallel to \( EF \). Point \( C \) is chosen in the interior of triangle \( AEF \) such that \( ABCD \) is cyclic. If \( BC = 3 \) and \( CD = 4 \), then the ratio of areas \(\frac{[ABCD]}{[AEF]}\) can be written as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).

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