
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, its right focus is $F(1,0)$, and the point $(-1, \dfrac{\sqrt{2}}{2})$ is on the ellipse $C$.

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

(Ⅱ) It is known that the moving line $l$ passes through point $F$ and intersects the ellipse $C$ at points $A$ and $B$. Is there a fixed point $Q$ on the $x$-axis such that $\overrightarrow{QA} \cdot \overrightarrow{QB} = -\dfrac{7}{16}$ always holds? If it exists, find the coordinates of point $Q$; if not, please explain why.

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