
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 described by the equation \[x^{2}+2y^{2}=m\ (m > 0)\], construct a chord \(AB\) with midpoint \(M(2,1)\) inside the ellipse. Suppose the perpendicular bisector of the chord \(AB\) intersects the ellipse at points \(C\) and \(D\).
\(\\(\)Ⅰ\(\\)\) Calculate the eccentricity of the ellipse.
\(\\(\)Ⅱ\(\\)\) Determine whether there exists a value of \(m\) such that points \(A\), \(B\), \(C\), and \(D\) lie on the same circle, and provide an explanation.

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