
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 scalene triangle with orthocenter  $H$  and circumcenter  $O$ . Let the line through  $A$  tangent to the circumcircle of triangle  $AHO$  intersect the circumcircle of triangle  $ABC$  at  $A$  and  $P \neq A$ . Let the circumcircles of triangles  $AOP$  and  $BHP$  intersect at  $P$  and  $Q \neq P$ . Let line  $PQ$  intersect segment  $BO$  at  $X$ . Suppose that  $BX=2$ ,  $OX=1$ , and  $BC=5$ . Then  $AB \cdot AC = \sqrt{k}+m\sqrt{n}$  for positive integers  $k$ ,  $m$ , and  $n$ , where neither  $k$  nor  $n$  is divisible by the square of any integer greater than  $1$ . Compute  $100k+10m+n$ .

*Proposed by Luke Robitaille*

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