
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 \(AB = 14\sqrt{3}\) and \(BC = 15\sqrt{3}\). Diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at \(P\). If triangle \(ABP\) is removed, and edges \(\overline{AP}\) and \(\overline{BP}\) are joined, followed by creasing along segments \(\overline{CP}\) and \(\overline{DP}\), a triangular pyramid is formed 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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