
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with eccentricity $e = \frac{\sqrt{3}}{2}$, and a line passing through the focus and perpendicular to the major axis intersects the ellipse at a segment of length $1$.

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

$(2)$ Let $D$, $E$, $F$ be three moving points on the ellipse $C$, with $D$ in the first quadrant, and $E$, $F$ symmetric about the origin, and $|DE| = |DF|$. Does the area of $\triangle DEF$ have a minimum value? If it exists, find the coordinates of point $D$; if not, explain why.

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