Volume of $n$-sphere contained in $\ell_1$ ball

For a given $ r>1$ , what is the surface area of $ \mathbb S^{n-1}$ (the sphere of radius 1 in $ \mathbb R^n$ ) which is contained outside of the $ \ell_1$ ball of radius $ r$ ? Or equivalently, if $ X\sim U(\mathbb S^{n-1})$ , a point sampled uniformly from the sphere, what is the probability that $ \Vert X\Vert_1\geq r$ ?

This is easy to compute for $ r\geq \sqrt{n-1}$ , as the area is exactly $ 2^n$ spherical caps, and this has a clean, closed-form formula. For smaller values of $ r$ , however, these caps intersect, and the algebra gets worse.

The exact value of this probability matters less than approximate asymptotic bounds for $ n$ large.