
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
Given a pyramid $P-ABCD$ with a rectangular base $ABCD$ where $AB = CD = 2$ and $BC = DA = 1$, the vertex $P$ is directly above the center of rectangle $ABCD$ and equidistant from all vertices $A, B, C, D$. Find the volume of the pyramid given that $\angle APB = \theta$.

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