
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 angle $θ$ between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, and $|\overrightarrow{a}|=4$, $|\overrightarrow{b}|=2$, compute the following:
$(1) \overrightarrow{a} \cdot \overrightarrow{b};$
$(2) (\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - 2\overrightarrow{b});$
$(3) |\overrightarrow{a} + \overrightarrow{b}|.$

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