
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 eccentricity $e = \dfrac{\sqrt{2}}{2}$, a circle with the top vertex and the right focus of the ellipse as the endpoints of its diameter is tangent to the line $x+y-2=0$.

(Ⅰ) Find the standard equation of the ellipse;

(Ⅱ) For the line $l$: $y=x+m$ and point $Q(0,3)$, determine if there exist two different points $A$ and $B$ on ellipse $C$ that are symmetric about line $l$ and satisfy $3\overrightarrow{QA} \cdot \overrightarrow{QB} = 32$. If such points exist, find the value of the real number $m$; if not, explain why.

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