
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
In  $\triangle ABC$ ,  $AB = 4$ ,  $BC = 5$ , and  $CA = 6$ . Circular arcs  $p$ ,  $q$ ,  $r$  of measure  $60^\circ$  are drawn from  $A$  to  $B$ , from  $A$  to  $C$ , and from  $B$  to  $C$ , respectively, so that  $p$ ,  $q$  lie completely outside  $\triangle ABC$  but  $r$  does not. Let  $X$ ,  $Y$ ,  $Z$  be the midpoints of  $p$ ,  $q$ ,  $r$ , respectively. If  $\sin \angle XZY = \dfrac{a\sqrt{b}+c}{d}$ , where  $a, b, c, d$  are positive integers,  $\gcd(a,c,d)=1$ , and  $b$  is not divisible by the square of a prime, compute  $a+b+c+d$ .

*Proposed by Michael Tang*

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