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

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