
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  $ABCD$  be a rectangle with  $AB=10$  and  $AD=5.$  Suppose points  $P$  and  $Q$  are on segments  $CD$  and  $BC,$  respectively, such that the following conditions hold:


- $BD \parallel PQ$
- $\angle APQ=90^{\circ}.$

 What is the area of  $\triangle CPQ?$ *Proposed by Kyle Lee*

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