
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 $C$ passing through point $A(0,1)$ with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, where its foci are $F_1$ and $F_2$ respectively. Let $B$ be any point on the ellipse, and $\sqrt{3}|BF_1|$, $|F_1F_2|$, $\sqrt{3}|BF_2|$ form an arithmetic sequence.
1. Find the standard equation of the ellipse $C$.
2. The line $l: y = k(x + 2)$ intersects the ellipse at points $P$ and $Q$. If point $A$ is always outside the circle with diameter $PQ$, find the range of the real number $k$.

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