
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 $|\vec{a}| = 2$, $|\vec{b}| = 1$, and $(2\vec{a} - 3\vec{b}) \cdot (2\vec{a} + \vec{b}) = 9$.

(I) Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$;

(II) Find $|\vec{a} + \vec{b}|$ and the projection of vector $\vec{a}$ in the direction of $\vec{a} + \vec{b}$.

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