
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
In the Cartesian coordinate system, circle $O$ intersects the $x$-axis at points $F_{1}$ and $F_{2}$, and intersects the $y$-axis at points $B_{1}$ and $B_{2}$. An ellipse $E$, with $B_{1}$ and $B_{2}$ as vertices and $F_{1}$, $F_{2}$ as the left and right foci respectively, exactly passes through the point $\left(1, \frac{\sqrt{2}}{2}\right)$.  
$(1)$ Find the standard equation of ellipse $E$;  
$(2)$ Suppose a line $l$ passing through the point $(-2,0)$ intersects ellipse $E$ at points $M$ and $N$, find the maximum area of $\triangle F_{2}MN$.

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