
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
For what values of the real number $m$, the points in the complex plane representing the complex number $z=(m^{2}-5m+6)+(m^{2}-3m+2)i$ satisfy the following conditions:<br/>$(1)$ coincide with the origin;<br/>$(2)$ lie on the line $y=2x$;<br/>$(3)$ lie in the first or third quadrant.

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