
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
The rectangle $ABCD$ has dimensions $AB = 15\sqrt{2}$ and $BC = 10\sqrt{2}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is removed and $\overline{AP}$ and $\overline{BP}$ are joined, creasing along segments $\overline{CP}$ and $\overline{DP}$, a triangular pyramid is formed. All four faces of this pyramid are isosceles triangles. Find the volume of this pyramid.

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