
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  $ABC$  is inscribed in a unit circle  $\omega$ . Let  $H$  be its orthocenter and  $D$  be the foot of the perpendicular from  $A$  to  $BC$ . Let  $\triangle XY Z$  be the triangle formed by drawing the tangents to  $\omega$  at  $A, B, C$ . If  $\overline{AH} = \overline{HD}$  and the side lengths of  $\triangle XY Z$  form an arithmetic sequence, the area of  $\triangle ABC$  can be expressed in the form  $\tfrac{p}{q}$  for relatively prime positive integers  $p, q$ . What is  $p + q$ ?


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