
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
Triangle  $ABC$  is scalene.  Points  $P$  and  $Q$  are on segment  $BC$  with  $P$  between  $B$  and  $Q$  such that  $BP=21$ ,  $PQ=35$ , and  $QC=100$ .  If  $AP$  and  $AQ$  trisect  $\angle A$ , then  $\tfrac{AB}{AC}$  can be written uniquely as  $\tfrac{p\sqrt q}r$ , where  $p$  and  $r$  are relatively prime positive integers and  $q$  is a positive integer not divisible by the square of any prime.  Determine  $p+q+r$ .

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