
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 $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ are in the same plane, and $\overrightarrow{a} = (1,2)$.  
$(1)$ If $|\overrightarrow{c}| = 3\sqrt{5}$ and $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$, find $\overrightarrow{c}$;  
$(2)$ If $|\overrightarrow{b}| = \sqrt{2}$ and $(\overrightarrow{a} + 2\overrightarrow{b}) \perp (\overrightarrow{a} - \overrightarrow{b})$, find the cosine value of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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