
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 the ellipse $C\_1: \frac{x^2}{8} + \frac{y^2}{4} = 1$ with left and right foci $F\_1$ and $F\_2$, a line $l\_1$ is drawn through point $F\_1$ perpendicular to the $x$-axis. Line $l\_2$ is perpendicular to $l\_1$ at point $P$, and the perpendicular bisector of segment $PF\_2$ intersects $l\_2$ at point $M$.

(I) Find the equation of the trajectory $C\_2$ of point $M$;
(II) Two perpendicular lines $AC$ and $BD$ are drawn through point $F\_2$, intersecting ellipse $C\_1$ at points $A$, $C$ and $B$, $D$ respectively. Find the minimum value of the area of quadrilateral $ABCD$.

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