Temperature of a sphere heating up in the sun (average emissivity of the sphere unknown)

I’m stuck at the following exercise:

Imagine an aluminum sphere that is painted matte white with an
emissivity of $epsilon=0.10$ at visible wavelengths and
$epsilon=0.95$ at wavelengths longer than $5text{ }mutext{m}$.
This sphere is orbiting the sun receiving a power flux of
$R_{sun}=1330text{ Wm}^{-2}$. Find the steady-state temperature of
the sphere. Find the peak wavelength of the radiated energy and
determine which value of $epsilon$ should be used to calculate the
radiated energy.

Now first I realized that the sphere with Radius $r$ only receives the radiation of the sun on an area of $pi r^2$ while being able to emmit radiation over its whole area of $4pi r^2$. So the power flux emitted by the sphere should be a quarter of the power flux it receives:

$$R_{sphere}=332.5text{ Wm}^{-2}$$

I don’t get much further than that though. I woud like to continue calculating the temperature, for example with $R=epsilon_{avg}sigma T^4$, but the average emissivity is dependent on the temperature and I don’t even know $epsilon$ for most wavelengths, so that’s not really an option. The only other relevant equation that I know is Planck’s law:

$$frac{dR}{dlambda}=frac{2pi hc^2}{lambda^5}frac{epsilon}{exp(hc/klambda T)-1}$$

But $epsilon$ is only given for visible light and wavelengths longer than 5 micrometers. I thought about approximating the gap between those with a linear function like this:

$$epsilon=left{begin{matrix}0.1&0<lambda<750nm.2lambda/mu m-0.05&750nm<lambda<5mu m.95&5mu m<lambdaend{matrix}right.$$

but even with that I don’t know how I could use Planck’s law to calculate the temperature. Any ideas are greatly appreciated.