
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:

$(1)$ $\vec{a} \cdot \vec{b}$  
$(2)$ $(2\vec{a} - \vec{b})(\vec{a} + 3\vec{b})$  
$(3)$ $|\vec{a} + \vec{b}|$

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