
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 $|\vec{a}|=2$, $|\vec{b}|=1$, and $(\vec{a}-\vec{b})\cdot(2\vec{a}+\vec{b})=8$.

$(1)$ Find the angle $\theta$ between $\vec{a}$ and $\vec{b}$;  
$(2)$ Find $|2\vec{a}-\vec{b}|$.

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