
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 $C$ with the equation $\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1,(a > b > 0)$ and an eccentricity of $\dfrac{1}{2}$, a circle with the radius equal to the length of the ellipse's minor axis is centered at the origin $O$ and is tangent to the line $x-y+\sqrt{6}=0$.

(1) Find the standard equation of the ellipse $C$.

(2) If line $l$ with equation $y=kx+m$ intersects ellipse $C$ at points $A$ and $B$, and $k_{OA} \cdot k_{OB}=-\dfrac{{{b}^{2}}}{{{a}^{2}}}$, determine whether the area of $\triangle AOB$ is a fixed value. If it is, find the fixed value; if not, explain why.

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