
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 $\vec{a}$ and $\vec{b}$ satisfying $|\vec{a}|=2$ and $|\vec{b}|=1$, with $\vec{AB} = 2\vec{a} - \vec{b}$ and $\vec{CD} = \vec{a} + 3\vec{b}$,
(1) If the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$, find the value of $|\vec{a} - \vec{b}|$;
(2) If $\vec{AB} \perp \vec{CD}$, find the value of the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$.

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