
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 \\(C\\): \\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)\\) with one vertex at \\((0, -1)\\) and an eccentricity \\(e= \frac{\sqrt{2}}{2}\\):
\\((1)\\) Find the equation of ellipse \\(C\\);
\\((2)\\) A line \\(L\\) passing through point \\(M(0,m) (-1 < m < 0)\\) intersects the ellipse \\(C\\) at points \\(A\\) and \\(B\\). Determine whether there exists a fixed point \\(T\\) on ellipse \\(C\\) such that any circle with diameter \\(AB\\) always passes through \\(T\\). If it exists, find the value of \\(m\\) and the coordinates of point \\(T\\); if not, explain why.

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