
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 the graph represented by the curve \( G \). The curve \( G \) has a focus at \( F \). A line \( l_1 \) passing through \( F \) intersects the curve \( G \) at points \( A \) and \( C \), and another line \( l_2 \) passing through \( F \) intersects the curve \( G \) at points \( B \) and \( D \). It is also given that \( \overrightarrow{AC} \cdot \overrightarrow{BD} = 0 \). 

(1) Find the equation of the curve \( G \) and the coordinates of the focus \( F \).
(2) Find the minimum value of the area \( S \) of the quadrilateral \( ABCD \).

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