On the Heads of Three Pins

A sphere of weight $100 \, \text{N}$ and radius $1 \, \text{m}$ has its center on the origin.

$\large{x = cos \theta \, sin \phi \\ y = sin \theta \, sin \phi \\ z = cos \phi }$

The sphere is supported by three thin vertical pins which make contact at $(\theta_1, \phi_1) = (0, \frac{2 \pi}{3})$ , $(\theta_2, \phi_2) = (\frac{\pi}{3}, \frac{3 \pi}{4})$ , and $(\theta_3, \phi_3) = (-\frac{4 \pi}{5}, \frac{5 \pi}{6})$ .

What is the sum of the magnitudes of the three contact forces?

Note: Gravity is in the negative $z$ direction. Contact forces are normal to the surface.