
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 $|\vec{a}|=4$ and $|\vec{b}|=3$.
$(1)$ If the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$, find $(\vec{a} +2\vec{b})\cdot(\vec{a} -3\vec{b})$;
$(2)$ If $(2\vec{a} -3\vec{b})\cdot(2\vec{a} +\vec{b})=61$, find the angle between $\vec{a}$ and $\vec{b}$.

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