
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
Given a point  $P$  on a triangular piece of paper  $ABC,$  consider the creases that are formed in the paper when  $A, B,$  and  $C$  are folded onto  $P.$   Let us call  $P$  a fold point of  $\triangle ABC$  if these creases, which number three unless  $P$  is one of the vertices, do not intersect.  Suppose that  $AB=36, AC=72,$  and  $\angle B=90^\circ.$   Then the area of the set of all fold points of  $\triangle ABC$  can be written in the form  $q\pi-r\sqrt{s},$  where  $q, r,$  and  $s$  are positive integers and  $s$  is not divisible by the square of any prime.  What is  $q+r+s$ ?

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