
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 $|\vec{a}|=1$, $\vec{a} \cdot \vec{b} = \frac{1}{2}$, and $(\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = \frac{1}{2}$, find:  
(1) The angle between $\vec{a}$ and $\vec{b}$.  
(2) The cosine of the angle between $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$.

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