
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
For rational numbers $a$, $b$, $n$, $d$, if $|a-n|+|b-n|=d$, then the "relative relationship value" of $a$ and $b$ with respect to $n$ is defined as $d$. For example: $|2-1|+|3-1|=3$, then the "relative relationship value" of $2$ and $3$ with respect to $1$ is $3$.  
$(1)$ The "relative relationship value" of $2$ and $-5$ with respect to $2$ is ______.  
$(2)$ If the "relative relationship value" of $m$ and $n$ with respect to $2$ is $2$, then the maximum value of $m+n$ is ______.

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