
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, a line $L$ passes through the upper and right vertices of the ellipse $C$ and is tangent to the circle $x^{2}+y^{2}= \frac{12}{7}$. Also, the right focus of the ellipse $C$ coincides with the focus of the parabola $y^{2}=4x$.
(I) Find the standard equation of the ellipse $C$;
(II) Two perpendicular rays are drawn from point $O$ and intersect the ellipse $C$ at points $A$ and $B$ (where $O$ is the origin of the coordinate system), find the minimum value of the area of $\triangle OAB$.

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