
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
Consider rectangle $ABCD$ with points $P$ and $Q$ such that $APD$ and $BQC$ are isosceles right triangles with right angles at $P$ and $Q$ respectively. How many right triangles can be drawn using any three of the points $\{A, P, B, C, Q, D\}$ as vertices? 
[asy]
draw((0,0)--(10,0)--(10,6)--(0,6)--cycle);
draw((0,3)--(5,6));
draw((10,3)--(5,0));
label("D",(0,0),S);
label("Q",(5,0),S);
label("C",(10,0),S);
label("B",(10,6),N);
label("P",(5,6),N);
label("A",(0,6),N);
[/asy]

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