
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 an ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{\sqrt{3}}{2}$, and the ellipse passes through the point $(0,1)$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ If the line $l_1$: $x+2y-2=0$ is tangent to the circle $D$: $x^2+y^2-6x-4y+m=0$:

$(i)$ Find the standard equation of the circle $D$;

$(ii)$ If the line $l_2$ passes through the fixed point $(3,0)$, intersects the ellipse $C$ at two distinct points $E$, $F$, and intersects the circle $D$ at two distinct points $M$, $N$, find the range of values for $|EF| \cdot |MN|$.

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