
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 vectors $|\vec{a}|=1$, $|\vec{b}|=2$.
- (I) If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$, find $|\vec{a}+2\vec{b}|$;
- (II) If $(2\vec{a}-\vec{b})\cdot(3\vec{a}+\vec{b})=3$, find the angle between $\vec{a}$ and $\vec{b}$.

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