
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 $\overrightarrow{a}=(1,3)$.<br/>$(1)$ If $|\overrightarrow{b}|=2\sqrt{10}$ and $\overrightarrow{a}$ is opposite to $\overrightarrow{b}$, find the coordinates of $\overrightarrow{b}$;<br/>$(2)$ If $|\overrightarrow{c}|=\sqrt{5}$ and $(\overrightarrow{a}+\overrightarrow{c})$ is perpendicular to $(2\overrightarrow{a}-3\overrightarrow{c})$, find the angle $\theta$ between $\overrightarrow{a}$ and $\overrightarrow{c}$.

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