Given a clock with hour, minute, and second hands that each move continuously (i.e., no “ticking” occurs), show whether there exists a time at which the lesser angle formed by each pair of the hands is $\frac{2 \pi}{3}$.

I ask this because, though I expect the solution to be fairly simple in retrospect, I have never before resolved a question of this form and do not know how to phrase any proof I had.