
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 $\overrightarrow{a}$ and $\overrightarrow{b}$ are plane vectors, and $\overrightarrow{a}=(1,2)$.  
$(1)$ If $\overrightarrow{b}=(2,1)$, and $k\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$, find the value of the real number $k$;  
$(2)$ If $\overrightarrow{a} \parallel \overrightarrow{b}$, and $|\overrightarrow{b}|=3\sqrt{5}$, find the coordinates of vector $\overrightarrow{b}$.

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