
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  $\triangle ABC$  be a triangle with side length  $BC= 4\sqrt{6}$ . Denote  $\omega$  as the circumcircle of  $\triangle{ABC}$ . Point  $D$  lies on  $\omega$  such that  $AD$  is the diameter of  $\omega$ . Let  $N$  be the midpoint of arc  $BC$  that contains  $A$ .  $H$  is the intersection of the altitudes in  $\triangle{ABC}$  and it is given that  $HN = HD= 6$ . If the area of  $\triangle{ABC}$  can be expressed as  $\frac{a\sqrt{b}}{c}$ , where  $a,b,c$  are positive integers with  $a$  and  $c$  relatively prime and  $b$  not divisible by the square of any prime, compute  $a+b+c$ .

*Proposed by Andy Xu*

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