
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, and a line passing through the left focus $F_1$ of the ellipse $C$ and perpendicular to the major axis intersects the ellipse $C$ at points $A$ and $B$, where $A(-1,-\dfrac{\sqrt{2}}{2})$.

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

$(2)$ If a circle $(x-x_0)^2+(y-y_0)^2=r^2$ passes through the right focus $F_2$ of the ellipse $C$, and intersects the $y$-axis at points $M$ and $N$, where $(x_0,y_0)$ is a point on the ellipse $C$, determine whether there exists a real number $x_0$ such that $\overrightarrow{F_2M} \cdot \overrightarrow{F_2N} = \dfrac{4}{3}$. If it exists, find the area of $\triangle F_2MN$; if not, explain why.

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