
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 three vectors on the plane $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, where $\overrightarrow{a}=(1,2)$.
$(1)$ If $|\overrightarrow{c}|=3\sqrt{5}$ and $\overrightarrow{a} \parallel \overrightarrow{c}$, find the coordinates of $\overrightarrow{c}$;
$(2)$ If $|\overrightarrow{b}|=3\sqrt{5}$ and $(4\overrightarrow{a}-\overrightarrow{b}) \perp (2\overrightarrow{a}+\overrightarrow{b})$, find the cosine of the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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