
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 a line _\(l\)_ with slope \(\sqrt{3}\) passing through the point \(\left(0,-2 \sqrt{3}\right)\) and the right focus of the ellipse _\(C\)_: \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)\), and the ellipse passes through the point \(\left(\sqrt{3},1\right)\).

(Ⅰ) Find the equation of the ellipse _\(C\)_.

(Ⅱ) Given the point _\(D\)_ \(\left(3,0\right)\), and points _\(M\)_ and _\(N\)_ are two distinct points on the ellipse _\(C\)_, and \(\overline{DM} : \overline{DN} = \lambda\), find the range of the real number _\(\lambda\)_.

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