Confusing square

Geometry Level 5

The rectangle \(ABCD^{}_{}\) below has dimensions \(AB^{}_{} = 12 \sqrt{3}\) and \(BC^{}_{} = 13 \sqrt{3}\). Diagonals \(\overline{AC}\) and \(\overline{BD}\) intersect at \(P^{}_{}\). If triangle \(ABP^{}_{}\) is cut out and removed, edges \(\overline{AP}\) and \(\overline{BP}\) are joined, and the figure is then creased along segments \(\overline{CP}\) and \(\overline{DP}\), we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.

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