
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 between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $120^{\circ}$, and $|\overrightarrow{a}|=3$, $|\overrightarrow{b}|=2$. Find:

1. $\overrightarrow{a} \cdot \overrightarrow{b}$;
2. $(2\overrightarrow{a}+\overrightarrow{b}) \cdot (\overrightarrow{a}-2\overrightarrow{b})$;
3. If $\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-k\overrightarrow{b}$, find the value of the real number $k$.

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