
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
A rectangle $ABCD$ has dimensions $AB = 15 \sqrt{2}$ and $BC = 17 \sqrt{2}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $P$. If we cut and remove triangle $ABP$ from the rectangle, and then join edges $\overline{AP}$ and $\overline{BP}$ while creasing along $\overline{CP}$ and $\overline{DP}$, it creates a triangular pyramid where all four faces are isosceles triangles. Find the volume of this pyramid.

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