
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 $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ lie in the same plane, and $\overrightarrow{a}=(-2,1)$.  
$(1)$ If $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$ and $|\overrightarrow{c}|=25$, find the coordinates of vector $\overrightarrow{c}$.  
$(2)$ If $\overrightarrow{b}=(3,2)$, and $k\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}+2\overrightarrow{b}$, find the value of $k$.

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