
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
Given a sequence $\{a_n\}$ satisfying $a_1=m$, $a_{n+1}=\begin{cases} 2a_n, & n=2k-1 \\ a_n+r, & n=2k \end{cases} (k\in\mathbb{N}^*, r\in\mathbb{R})$, and its sum of the first $n$ terms is $S_n$. If for any $n\in\mathbb{N}^*$, the sequence $\{a_n\}$ satisfies $a_{n+2}=a_n$, then the relationship between $m$ and $r$ is \_\_\_\_.

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