
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 a vertex coordinate at $B(0,1)$, and the eccentricity of the ellipse being $\frac{\sqrt{3}}{2}$,
$(I)$ find the equation of the ellipse;
$(II)$ let point $Q$ be a point on ellipse $C$ below the x-axis, and $F_1$, $F_2$ are the left and right foci of the ellipse, respectively. If the slope of line $QF_1$ is $\frac{\pi}{6}$, find the area of $\triangle QF_1F_2$.

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