
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
Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle such that chords  $\overline{AC}$  and  $\overline{BD}$  intersect at a point  $E$  inside the circle.  Suppose that  $\angle ADE =\angle CBE = 75^\circ$ ,  $BE=4$ , and  $DE=8$ .  The value of  $AB^2$  can be written in the form  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$ , and  $c$  such that  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Tony Kim*

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