
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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

Present the answer in LaTex format: \boxed{Your answer}
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