
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 the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) passes through the point $M(1, \dfrac{\sqrt{3}}{2})$, $F_1$, $F_2$ are the two foci of the ellipse $C$, $|F_1F_2| = 2\sqrt{3}$, $P$ is a moving point on the ellipse $C$.

(Ⅰ) Find the standard equation of the ellipse $C$;

(Ⅱ) If point $P$ is in the first quadrant, and $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} \leqslant \dfrac{1}{4}$, find the range of the $x$-coordinate of point $P$;

(Ⅲ) Is there a line $l$ passing through the fixed point $N(0,2)$ that intersects the ellipse $C$ at two distinct points $A$, $B$, making $\angle AOB = 90^{\circ}$ (where $O$ is the origin)? If it exists, find the slope $k$ of line $l$; if not, explain why.

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