
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 a circle $C$: $x^{2}+y^{2}+Dx+Ey+3=0$ is symmetric about the line $x+y-1=0$, and the center of circle $C$ is in the fourth quadrant with a radius of $\sqrt{2}$.
(Ⅰ) Find the equation of circle $C$;
(Ⅱ) Is there a line $l$ that is tangent to circle $C$ and has an x-intercept that is twice its y-intercept? If it exists, find the equation of line $l$; if not, explain why.

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