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

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