How many three-digit positive integers $\underline{a}\underline{b}\underline{c}$ are there whose nonzero digits $a,b,$ and $c$ satisfy $0.\overline{\underline{a}\underline{b}\underline{c}}=\frac{1}{3}(2\cdot 0.\overline{\underline{a}}+0.\overline{\underline{b}}+0.\overline{\underline{c}})$?
(The bar indicates repetition, thus $0.\overline{\underline{a}\underline{b}\underline{c}}$ is the infinite repeating decimal $0.\underline{a}\underline{b}\underline{c}\underline{a}\underline{b}\underline{c}\cdots$)