
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
In the plane rectangular coordinate system \( xOy \), the ellipse \( C: \frac{x^2}{3} + y^2 = 1 \) has its upper vertex at \( A \). A line \( l \) that does not pass through point \( A \) intersects the ellipse \( C \) at points \( P \) and \( Q \), and \(\overrightarrow{AP} \cdot \overrightarrow{AQ} = 0\).

1. Does the line \( l \) pass through a fixed point? If so, find the coordinates of the fixed point; if not, explain why not.
2. Tangents to the ellipse are drawn at points \( P \) and \( Q \), intersecting at point \( B \). Determine the range of the area of triangle \( \triangle BPQ \).

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