
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
The eccentricity of the ellipse $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $\frac{\sqrt{2}}{2}$, with left and right foci $F_{1}$ and $F_{2}$ respectively, and the top vertex $A$. The distance from point $F_{1}$ to the line $AF_{2}$ is $\sqrt{2}$.  
$(1)$ Find the equation of $C_{1}$.  
$(2)$ A line $l$ passing through point $Q(\sqrt{3},0)$ intersects the right branch of the hyperbola $C_2: x^2 - y^2 = 1$ at points $M$ and $N$, where point $P$ lies on $C_{1}$. Find the range of values for the area of $\triangle PMN$.

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