
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  $ABC$  be an acute triangle with circumcircle  $\omega$ . Let  $D$  and  $E$  be the feet of the altitudes from  $B$  and  $C$  onto sides  $AC$  and  $AB$ , respectively. Lines  $BD$  and  $CE$  intersect  $\omega$  again at points  $P \neq B$  and  $Q \neq C$ . Suppose that  $PD=3$ ,  $QE=2$ , and  $AP \parallel BC$ . Compute  $DE$ .

*Proposed by Kyle Lee*

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