
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 that $a_{1}, a_{2}, \cdots, a_{10}$ are ten different positive integers satisfying  the equation $\left|a_{i+1}-a_{i}\right|=2 \text { or } 3$, where $i=1,2, \cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1}, a_{2}, \cdots, a_{10}$ and $m$ is the minimum number among $a_{1}, a_{2}, \cdots, a_{10}$.

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