
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 that the eccentricity of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ is $\frac{1}{2}$, the hyperbola equation is $\frac{y^{2}}{a^{2}}- \frac{x^{2}}{b^{2}}=1 (a > 0, b > 0)$, and the distance between the intersection points of the line $x=2$ and the hyperbola is $\frac{4\sqrt{21}}{3}$.
(I) Find the equations of the ellipse and hyperbola.
(II) A line $l$ passing through the focus $F_{2}$ intersects the ellipse at points $M$ and $N$, and intersects the hyperbola at points $P$ and $Q$. When the area of the incircle of triangle $F_{1}MN$ ($F_{1}$ being the left focus of the ellipse) reaches its maximum value, find the area of triangle $F_{1}PQ$.

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