Bending A Triangle

Geometry Level 3

Triangle \( ABC\) has coodinates \(A= (-4, 0)\), \(B= (4 , 0)\), and \(C= (0 , 3)\).

Let \(P\) be the point in the first quadrant such that \(\triangle ABP\) has half the area of \(\triangle ABC\) but both triangles have the same perimeter.

What is the length of \(CP\)? If your solution is in a form of \(\sqrt{d}\), submit \(d\) as the answer.

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