
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) that passes through point A(0,1), with F<sub>1</sub> and F<sub>2</sub> as its left and right foci, respectively. Point B is any point on the ellipse such that the lengths of $\sqrt{3}|BF_1|$, $|F_1F_2|$, and $\sqrt{3}|BF_2|$ form an arithmetic sequence.
(1) Find the standard equation of the ellipse C.
(2) A 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 values for the real number k.

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