
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 an acute scalene triangle  $ABC$ , points  $D,E,F$  lie on sides  $BC, CA, AB$ , respectively, such that  $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes  $AD, BE, CF$  meet at orthocenter  $H$ . Points  $P$  and  $Q$  lie on segment  $EF$  such that  $AP \perp EF$  and  $HQ \perp EF$ . Lines  $DP$  and  $QH$  intersect at point  $R$ . Compute  $HQ/HR$ .

*Proposed by Zuming Feng*

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