
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 complex numbers $z_{1}=m(m-1)+(m-1)i$ and $z_{2}=(m+1)+(m^{2}-1)i$ ($m\in\mathbb{R}$), the corresponding points in the complex plane are $z_{1}$ and $z_{2}$ respectively.  
$(1)$ If $z_{1}$ is a pure imaginary number, find the value of $m$;  
$(2)$ If the point corresponding to $z_{2}$ in the complex plane is located in the fourth quadrant, find the range of values for $m$.

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