
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 a triangle with incenter  $I$ . Let  $P$  and  $Q$  be points such that  $IP\perp AC$ ,  $IQ\perp AB$ , and  $IA\perp PQ$ . Assume that  $BP$  and  $CQ$  intersect at the point  $R\neq A$  on the circumcircle of  $ABC$  such that  $AR\parallel BC$ . Given that  $\angle B-\angle C=36^\circ$ , the value of  $\cos A$  can be expressed in the form  $\frac{m-\sqrt n}{p}$  for positive integers  $m,n,p$  and where  $n$  is not divisible by the square of any prime. Find the value of  $100m+10n+p$ .

*Proposed by Michael Ren*

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