
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its foci at $F_1$ and $F_2$, and the point $A$ has coordinates $(0,b)$. The line $AF_2$ intersects the ellipse at another point $B$. The perimeter of $\Delta AF_1B$ is $4\sqrt{3}$, the distance from $F_1$ to the line $AF_2$ is $\dfrac{2\sqrt{6}}{3}$, and the eccentricity of the ellipse $e > \dfrac{\sqrt{3}}{3}$,

(1) Find the equation of the ellipse;

(2) Given a fixed point $E(1,0)$, if the line $y=kx+2$ intersects the ellipse at points $P$ and $Q$. Is there a value of $k$ such that the circle with diameter $PQ$ passes through point $E$? If so, please explain the reason and find the value of $k$.

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