
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  $ABC$  has side lengths  $AB=18$ ,  $BC=36$ , and  $CA=24$ . The circle  $\Gamma$  passes through point  $C$  and is tangent to segment  $AB$  at point  $A$ . 
Let  $X$ , distinct from  $C$ , be the second intersection of  $\Gamma$  with  $BC$ . Moreover, let  $Y$  be the point on  $\Gamma$  such that segment  $AY$  is an angle bisector of  $\angle XAC$ .
Suppose the length of segment  $AY$  can be written in the form  $AY=\frac{p\sqrt{r}}{q}$  where  $p$ ,  $q$ , and  $r$  are positive integers such that  $gcd(p, q)=1$  and  $r$  is square free.
Find the value of  $p+q+r$ .

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