
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 $\overrightarrow{a}=(2,-1), \overrightarrow{b}=(1,2), \overrightarrow{c}=(3,-4)$, find:<br/>
$(1)$ If $|\overrightarrow{e}|=1$ and $\overrightarrow{e}∥\overrightarrow{a}$, find the coordinates of $\overrightarrow{e}$;<br/>
$(2)$ If $\overrightarrow{c}=m\overrightarrow{a}+n\overrightarrow{b}$, find $m+n$;<br/>
$(3)$ If $(k\overrightarrow{a}+\overrightarrow{b})⊥\overrightarrow{c}$, find the value of $k$.

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