Bend and Circulate

Consider the following vector field:

$\large{\vec{F} = (F_x,F_y,F_z) = (-y, x, 0) }$

Construct a curve according to the following directions:

1) Start with a unit-circle in the $xy$ plane with its center on the origin
2) Keep points $(0,-1)$ and $(0,1)$ pinned to the $xy$ plane so they can't move
3) Bend the right $(x>0)$ half of the circle toward the $+z$ direction so that it makes an angle of $\frac{\pi}{4}$ with the $xy$ plane. The points on the right side of the circle should still be co-planar after the bend.
4) Bend the left $(x<0)$ half of the circle toward the $+z$ direction so that it makes an angle of $\frac{\pi}{4}$ with the $xy$ plane. The points on the left side of the circle should still be co-planar after the bend.

The two bending operations should yield a closed and continuous (but not continuously differentiable) curve. Determine the absolute value of the circulation of $\vec{F}$ over the curve.