BALKAN 2006 Problem 4

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$ , and $$a_{n+1} = \begin{cases} \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, a_n + m & \textrm{ otherwise. } \end{cases}$$ Find all values of $a$ such that the sequence is periodical (starting from the beginning).