
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 $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are three vectors in the same plane, where $\overrightarrow{a} = (1,2)$.  
$(1)$ If $|\overrightarrow{b}| = 2\sqrt{5}$ and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the coordinates of $\overrightarrow{b}$.  
$(2)$ If $|\overrightarrow{c}| = \sqrt{10}$ and $2\overrightarrow{a} + \overrightarrow{c}$ is perpendicular to $4\overrightarrow{a} - 3\overrightarrow{c}$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{c}$.

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