
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 Cartesian coordinate plane $xOy$, the ellipse $C: \frac{x^2}{3} + y^2 = 1$ has its upper vertex at $A$. A line $l$ intersects the ellipse $C$ at points $P$ and $Q$, but does not pass through point $A$, 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 the reason.
(2) Tangent lines to the ellipse are drawn at points $P$ and $Q$. These tangents intersect at point $B$. Determine the range of the area of $\triangle BPQ$.

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