
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 square $ABCD$ with side length $2$. A point $P$ is marked on side $AB$, $Q$ on $BC$, and $R$ on $CD$, with $AP = 1$ and $CQ = 1.5$. These points are connected to the center of the square, $O$, forming trapezoids $APRO$, $OCQR$, and a pentagon $BQROP$. Calculate the length of segment $PQ$, and determine the area of trapezoid $APRO$. Assume $OP = OQ = OR = 1$.
**A)** Length of $PQ = 0.5$, Area of $APRO = 1$  
**B)** Length of $PQ = 0.5$, Area of $APRO = 0.75$  
**C)** Length of $PQ = 1$, Area of $APRO = 1$  
**D)** Length of $PQ = 1.5$, Area of $APRO = 0.5$

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