
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with eccentricity $e = \frac{\sqrt{2}}{2}$, a circle whose diameter is determined by the top vertex of the ellipse and its right focus, is tangent to the line $x + y - 2 = 0$.  
(1) Find the standard equation of the ellipse.  
(2) For the line $l: y = x + m$ and the point $Q(0,3)$, is there two different points $A$ and $B$ on ellipse $C$ such that they are symmetric with respect to line $l$ and $3\vec{QA} \cdot \vec{QB} = 32$? Provide the value of $m$ if it exists, otherwise explain why it does not.

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