
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 the vector $|\overrightarrow{m}|=2$ and $\overrightarrow{n}$ is a unit vector, with the angle between $\overrightarrow{m}$ and $\overrightarrow{n}$ being $60^{\circ}$, find:<br/>
$(1)$ $\overrightarrow{m} \cdot \overrightarrow{n}$;<br/>
$(2)$ $|\overrightarrow{m}+2\overrightarrow{n}|$.

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