
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, one endpoint of its minor axis and its two foci form an equilateral triangle with an area of $\sqrt{3}$. A line $l$ with slope $k (k \neq 0)$ passing through the right focus of the ellipse $C$ intersects the ellipse at points $A$ and $B$, and the midpoint of the line segment $AB$ is $P$.

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

$(2)$ A line passing through point $P$ and perpendicular to $AB$ intersects the $x$-axis at point $D$. Find the range of values for $\frac{|DP|}{|AB|}$.

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