
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 a rectangle $ABCD$ where segment $PQ$ bisects side $AD$ at point $P$ and side $BC$ at point $Q$. Point $R$ is placed on segment $PQ$ such that triangle $APR$ is a right triangle with the right angle at $P$. How many right triangles can be drawn using three of the points $\{A, B, C, D, P, Q, R\}$ as vertices?
[asy]
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle);
draw((0,2)--(8,2));
label("D",(0,0),S);
label("P",(0,2),W);
label("A",(0,4),N);
label("B",(8,4),N);
label("Q",(8,2),E);
label("C",(8,0),S);
dot((4,2));
label("R",(4,2),N);
[/asy]

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