
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
Consider the ellipse $C\_1$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its right focus coinciding with the focus of the parabola $C\_2$: $y^{2}=2px (p > 0)$. The curves $C\_1$ and $C\_2$ intersect at the point $(\frac{2}{3}, \frac{2}{3}\sqrt{6})$.
(I) Find the equation of the ellipse $C\_1$;
(II) A line $l$ passing through the right focus $F\_2$ (not parallel to the $x$-axis) intersects the ellipse $C\_1$ at points $A$ and $C$. Let $G$ be the midpoint of the line segment $AC$. The line $OG$ is extended to intersect the ellipse $C\_1$ at point $B$ ($O$ is the coordinate origin). Find the minimum value of the area $S$ of the quadrilateral $OABC$.

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