
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 $\vec{a}$, $\vec{b}$, and $\vec{c}$ are three vectors in the same plane, where $\vec{a}=(1,2)$, $\vec{b}=(-2,3)$, $\vec{c}=(-2,m)$  
$(1)$ If $\vec{a}\perp(\vec{b}+\vec{c})$, find the value of $m$;  
$(2)$ If $k\vec{a}+\vec{b}$ is collinear with $2\vec{a}-\vec{b}$, find the value of $k$.

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