
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 plane vectors $\overrightarrow{a}=(1,3)$, $\overrightarrow{b}=(2x-1,-x)$ where $x\in Z$.
$(1)$ If $2\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-2\overrightarrow{b}$, find $x$;
$(2)$ If vector $\overrightarrow{c}=(7,-1)$, and $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{b}-\overrightarrow{c}$, find $|\overrightarrow{a}-\overrightarrow{b}|$.

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