
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  $\angle B - \angle C = 30^{\circ}$ . Let  $D$  be the point where the  $A$ -excircle touches line  $BC$ ,  $O$  the circumcenter of triangle  $ABC$ , and  $X,Y$  the intersections of the altitude from  $A$  with the incircle with  $X$  in between  $A$  and  $Y$ . Suppose points  $A$ ,  $O$  and  $D$  are collinear. If the ratio  $\frac{AO}{AX}$  can be expressed in the form  $\frac{a+b\sqrt{c}}{d}$  for positive integers  $a,b,c,d$  with  $\gcd(a,b,d)=1$  and  $c$  not divisible by the square of any prime, find  $a+b+c+d$ .

*James Tao*

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