Find a harmonic function on $|z|<1$ with value 1 on an arc of $|z|=1$ and zero on the rest.

Of course, if allow the arc with value $1$ to be the whole $|z|=1$ circle it would be trivial.

Without that, how to construct such a function, I'm a bit lost. I suppose if I can find a holomorphic function $f(z)$ whose real part full fill that it shall work, but playing around with the normal functions such as $e^x$, $\sin x$ etc, I can't find one with such property.