Don't let the quantifiers scare you

Let $P((x,y),(x_0,y_0),z,\epsilon,\delta)$ be the statment that $(\cos(x_0\cos^{-1}(z))+\cos(y_0\cos^{-1}(z))=1+z) \wedge (||(x,y)-(x_0,y_0)||<\epsilon)$ . Let $S = \{ (x,y) \in \mathbb{R}^2 : (\forall \epsilon > 0)(\exists \delta \in (0,1))(\forall z:1-\delta<z<1)(\exists (x_0,y_0) \in \mathbb{R}^2: |x_0|,|y_0| \leq 1)(P((x,y),(x_0,y_0),z,\epsilon,\delta)) \}$ Then $S$ is in fact a 1-manifold of finite length. What is the length of $S$ ?