
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 $AB = a$, $BC = b$, and $CD = a$. Let the vertex $P$ of the pyramid be equidistant from the vertices $A, B, C, D$, and let $\angle APB = 2\theta$. Given that $a^2 + b^2 = 2$, find the volume of the pyramid.

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