
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
Let the equation of ellipse $E$ be $\frac{x^2}{a^2} + y^2 = 1 (a > 1)$, $O$ be the origin, and line $l$ intersects ellipse $E$ at points $A$ and $B$. Let $M$ be the midpoint of segment $AB$.
$(1)$ If $A$ and $B$ are respectively the left and top vertices of $E$, and the slope of $OM$ is $- \frac{1}{2}$, find the standard equation of $E$;
$(2)$ If $a=2$ and $|OM|=1$, find the maximum area of $\triangle AOB$.

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