
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{1}{2}$, let $A(0,b)$, $B(a,0)$, $F_1$, and $F_2$ be the left and right foci of the ellipse, respectively, and the area of $\triangle ABF_2$ is $\frac{\sqrt{3}}{2}$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ A line passing through $F_1$ intersects with a parabola that has $F_2$ as its focus and its vertex at the origin at points $P$ and $Q$. If $\overrightarrow{F_1P} = \lambda \overrightarrow{F_1Q}$ and $\lambda \in [2,3]$, find the range of the area of $\triangle F_2PQ$.

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