
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. $(\overrightarrow{a}-2\overrightarrow{b})\cdot(\overrightarrow{a}+ \overrightarrow{b})$;
2. The projection of $\overrightarrow{a}$ on $\overrightarrow{b}$;
3. The angle between $\overrightarrow{a}$ and $\overrightarrow{a}+ \overrightarrow{b}$.

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