
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

On a paper with a Cartesian coordinate system, draw a circle with center $A(-1,0)$ and radius $2\sqrt{2}$, with a fixed point $B(1,0)$. Fold the paper so that a point $P$ on the circumference of the circle coincides with point $B$, and connect $A$ to $P$ with the fold line intersecting at point $C$. Repeatedly folding this way creates a locus of moving point $C$.

(1) Find the equation of the curve $E$ corresponding to the locus of point $C$;

(2) If points $S$, $T$, $M$, and $N$ lie on the curve $E$ such that vectors $\overrightarrow{S B}$ and $\overrightarrow{B T}$ are collinear, vectors $\overrightarrow{M B}$ and $\overrightarrow{B N}$ are collinear, and $\overrightarrow{S B} \cdot \overrightarrow{M B} = 0$, find the minimum and maximum values of the area of quadrilateral $S M T N$.

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