
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 the 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 purely imaginary number, find the value of the real number $x$;

(2) In the complex plane, if the point corresponding to ${z_{1}}$ lies in the fourth quadrant and the point corresponding to ${z_{2}}$ lies 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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