Starting with the vertices \[P_1=(0,1),\ P_2=(1,1),\ P_3=(1,0),\ P_4=(0,0)\] of a square, we construct further points as follows:

- \(P_5\) is the midpoint of \(P_1P_2\).
- \(P_6\) is the midpoint of \(P_2P_3\).
- \(P_7\) is the midpoint of \(P_3P_4,\) and so on.

The spiral path approaches a point \(P_\infty=\left(\dfrac{A}{C},\dfrac{B}{C}\right)\) inside the square, where \(A, B, C\) are coprime positive integers.

Find \(A+B+C.\)

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