
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  $\tfrac{[ABCD]}{[AEF]}$  can be written as  $\tfrac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

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