
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 $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ $(a > b > 0)$ with eccentricity $\frac {\sqrt {2}}{2}$, the length of the line segment cut from the ellipse by a line passing through the right focus $F$ and perpendicular to the major axis is $\sqrt {2}$. Let $O$ be the origin of coordinates.  
$(1)$ Find the standard equation of the ellipse $C$;  
$(2)$ Suppose a line $l$ passes through point $M(0,2)$ and intersects the ellipse $C$ at points $A$ and $B$. Determine the maximum area of $\triangle AOB$ and the equation of the corresponding line $l$.

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