
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 $\vec{a}, \vec{b}$ satisfy $|\vec{a}| =2$, $|\vec{b}| =1$, and vectors $\vec{AB\;}=2 \vec{a}- \vec{b}$, $\vec{CD\;}= \vec{a}+3 \vec{b}$.  
$(1)$ If the angle between $\vec{a}, \vec{b}$ 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 $\vec{a}, \vec{b}$.

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