
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
It is given an acute triangle   $ABC$  ,  $AB \neq AC$   where the feet of altitude from  $A$  its  $H$ .  In the extensions of the sides  $AB$  and  $AC$   (in the direction of  $B$  and  $C$ ) we take the points  $P$  and  $Q$  respectively such that  $HP=HQ$  and the points  $B,C,P,Q$  are concyclic.
Find the ratio  $\tfrac{HP}{HA}$ .

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