
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 $z_{1}=x^{2}-1+(x^{2}-3x+2)i$ and $z_{2}=x+(3-2x)i$, where $x\in\mathbb{R}$.

$(1)$ If $z_{1}$ is a pure imaginary number, find the value of the real number $x$;  
$(2)$ In the complex plane, if the point corresponding to $z_{1}$ is in the fourth quadrant and the point corresponding to $z_{2}$ is in the first quadrant, find the range of values for the real number $x$.

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