
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 the equation in terms of $x$: $\frac{3x-m}{2}-\frac{x+m}{3}=\frac{5}{6}$.  
$(1)$ If $m=-1$, find the solution to the equation.  
$(2)$ If $x=5$ is a solution to the equation, find the value of $\frac{1}{2}m^2+2m$.

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