
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, one vertex is $B(0,1)$. The equation of ellipse $C$ is ______, and $P$, $Q$ are any two points on ellipse $C$ other than point $B$, and $BP \perp BQ$. The line $PQ$ passes through the fixed point ______.

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