
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 $|\vec{a}|=2$, $|\vec{b}|=3$, and the angle between $|\vec{a}|$ and $|\vec{b}|$ is $120^{\circ}$, find
$(1) \vec{a}\cdot \vec{b}$
$(2) \vec{a}^{2}- \vec{b}^{2}$
$(3)(2\vec{a}- \vec{b})( \vec{a}+3\vec{b})$
$(4)| \vec{a}+ \vec{b}|$

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