
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
In acute triangle  $\triangle ABC$ , point  $R$  lies on the perpendicular bisector of  $AC$  such that  $\overline{CA}$  bisects  $\angle BAR$ . Let  $Q$  be the intersection of lines  $AC$  and  $BR$ . The circumcircle of  $\triangle ARC$  intersects segment  $\overline{AB}$  at  $P\neq A$ , with  $AP=1$ ,  $PB=5$ , and  $AQ=2$ . Compute  $AR$ .

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