
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 function \( y = \sqrt{2x^2 + 2} \) with its graph represented as curve \( G \), and the focus of curve \( G \) denoted as \( F \), two lines \( l_1 \) and \( l_2 \) pass through \( F \) and intersect curve \( G \) at points \( A, C \) and \( B, D \) respectively, such that \( \overrightarrow{AC} \cdot \overrightarrow{BD} = 0 \).

(1) Find the equation of curve \( G \) and the coordinates of its focus \( F \).

(2) Determine the minimum value of the area \( S \) of quadrilateral \( ABCD \).

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