Spiralling out of control!

Starting with the vertices $P_1(0, 1), P_2(1, 1), P_3(1, 0)$ and $P_4(0, 0)$ of a square, we construct further points as shown in the figure: $P_5$ 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 polygonal spiral path $P_1P_2P_3P_4P_5P_6P_7$ ..... approaches a point $P$ inside the square.

Find and submit the distance of the point $P$ from the origin.

Courtesy: Stewart Calculus Sixth Edition