
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 focal points $F_{1}$ and $F_{2}$, a line $l_{1}$ is drawn through point $F_{1}$ perpendicular to the x-axis. A line $l_{2}$ intersects $l_{1}$ perpendicularly at point $P$. The perpendicular bisector of the line segment $PF_{2}$ intersects $l_{2}$ at point $M$.

1. Find the equation of the locus $C_{2}$ of point $M$.
2. Draw two mutually perpendicular lines $AC$ and $BD$ through point $F_{2}$, intersecting the ellipse at points $A$, $B$, $C$, and $D$. Find the minimum value of the area of the quadrilateral $ABCD$.

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