
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 $z\_1=2x+1+(x^2-3x+2)i, z\_2=x^2-2+(x^2+x-6)i (x \in \mathbb{R})$,

1. Find the value or range of the real number $x$ if $z\_1$ is a purely imaginary number.
2. Find the value or range of the real number $x$ if $z\_1 > z\_2$.

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