
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 line $l$: $y=2x+m$, and the ellipse $C$: $\dfrac{x^{2}}{4}+ \dfrac{y^{2}}{2}=1$. Determine the value of $m$ for which the line $l$ and the ellipse $C$:

$(1)$ have two distinct intersection points;

$(2)$ have exactly one intersection point;

$(3)$ have no intersection points.

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