
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 plane rectangular coordinate system $xOy$, given the ellipse $C$: $\frac{{x}^{2}}{4}+{y}^{2}=1$, the line $l:y=x+t(t$ is a real number and $t\neq 0)$ intersects the ellipse $C$ at points $A$ and $B$. <br/>$(1)$ If the line $l$ passes through the right focus of the ellipse, find the area of $\triangle OAB$; <br/>$(2)$ Let $M$ be the midpoint of the line segment $AB$, find the slope of the line $OM$.

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