
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 vectors $\overrightarrow{a}=(-3,1)$, $\overrightarrow{b}=(1,-2)$, $\overrightarrow{m}=\overrightarrow{a}+k\overrightarrow{b}$ ($k\in \mathbb{R}$).  
$(1)$ If $\overrightarrow{m}$ is perpendicular to the vector $2\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.  
$(2)$ If the vector $\overrightarrow{c}=(1,-1)$, and $\overrightarrow{m}$ is parallel to the vector $k\overrightarrow{b}+\overrightarrow{c}$, find the value of the real number $k$.

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