
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 $\frac{1}{2}$, a circle $\odot E$ with center at the origin and radius equal to the minor axis of the ellipse is tangent to the line $x-y+\sqrt{6}=0$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ A line passing through the fixed point $Q(1,0)$ with slope $k$ intersects the ellipse $C$ at points $M$ and $N$. If $\overrightarrow{OM}•\overrightarrow{ON}=-2$, find the value of the real number $k$ and the area of $\triangle MON$.

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