The Tale of Point P

Geometry Level 2

The \(\triangle AOB\) consists of point \(A = (0 , 1)\), point \(O = (0 , 0)\), and point \(B\) lying somewhere on the \(x\)-axis.

Let \(P\) be the point in the first quadrant such that \(AP = PB\) and \(AO \parallel PB\), as shown above.

If the length of \(OP\) is \(\sqrt{194}\), what is the area of the quadrilateral \( AOBP? \)

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