
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 $\overrightarrow {a}$, $\overrightarrow {b}$, and $\overrightarrow {c}$ in a plane, where $\overrightarrow {a}=(1,3)$.
1. If $|\overrightarrow {c}|=2 \sqrt {10}$ and $\overrightarrow {c}$ is parallel to $\overrightarrow {a}$, find $\overrightarrow {c}$ and the dot product of $\overrightarrow {a}$ and $\overrightarrow {c}$.
2. If $|\overrightarrow {b}|= \frac{\sqrt {10}}{2}$ and $\overrightarrow {a}-3 \overrightarrow {b}$ is perpendicular to $2 \overrightarrow {a}+ \overrightarrow {b}$, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.

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