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

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