
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  $\vartriangle ABC$  be a triangle. Let  $Q$  be a point in the interior of  $\vartriangle ABC$ , and let  $X, Y,Z$  denote the feet of the altitudes from  $Q$  to sides  $BC$ ,  $CA$ ,  $AB$ , respectively. Suppose that  $BC = 15$ ,  $\angle ABC = 60^o$ ,  $BZ = 8$ ,  $ZQ = 6$ , and   $\angle QCA = 30^o$ . Let line  $QX$  intersect the circumcircle of   $\vartriangle XY Z$  at the point  $W\ne  X$ . If the ratio  $\frac{ WY}{WZ}$  can be expressed as  $\frac{p}{q}$  for relatively prime positive integers  $p, q$ , find  $p + q$ .

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