
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  $ABC$  be an acute triangle with  $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$ , and circumradius  $72$ . Let  $ABC$  have circumcenter  $O$ , symmedian point  $K$ , and nine-point center  $N$ . Consider all non-degenerate hyperbolas  $\mathcal H$  with perpendicular asymptotes passing through  $A,B,C$ . Of these  $\mathcal H$ , exactly one has the property that there exists a point  $P\in \mathcal H$  such that  $NP$  is tangent to  $\mathcal H$  and  $P\in OK$ . Let  $N'$  be the reflection of  $N$  over  $BC$ . If  $AK$  meets  $PN'$  at  $Q$ , then the length of  $PQ$  can be expressed in the form  $a+b\sqrt{c}$ , where  $a,b,c$  are positive integers such that  $c$  is not divisible by the square of any prime. Compute  $100a+b+c$ .

*Proposed by Vincent Huang*

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