
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 $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three vectors in the same plane, where $\overrightarrow{a}=(1,-2)$.  
$(1)$ If $|\overrightarrow{c}|=2\sqrt{5}$ and $\overrightarrow{c}∥\overrightarrow{a}$, find the coordinates of $\overrightarrow{c}$;  
$(2)$ If $|\overrightarrow{b}|=1$ and $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-2\overrightarrow{b}$, find the cosine value of the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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