
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 $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a > b > 0)$, the equation of the incircle of the quadrilateral formed by connecting the four vertices of the ellipse in order is $({x}^{2}+{y}^{2}=\frac{12}{7})$, and the area of the quadrilateral is $4\sqrt{3}$.

(I) Find the equation of the ellipse $C$;

(II) Two mutually perpendicular rays are drawn from the origin $O$ intersecting the ellipse $C$ at points $A$ and $B$, respectively. Find the minimum value of the area of $\mathrm{\Delta }OAB$.

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