
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 two lines $l_{1}$: $x+m^{2}y+6=0$ and $l_{2}$: $\left(m-2\right)x+3my+2m=0$, for what values of $m$ do $l_{1}$ and $l_{2}$:
- $(1)$ intersect;
- $(2)$ coincide;
- $(3)$ exist a fixed point on line $l_{2}$? If it exists, find the point. If not, explain the reason.

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