
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 in the Cartesian coordinate system, \\(O\\) is the origin, and a moving circle \\(P\\) passes through the point \\(F(0,1)\\) and is tangent to the line \\(l:y=-1\\).

\\((1)\\) Find the equation of the trajectory \\(C\\) of the center \\(P\\) of the moving circle;

\\((2)\\) A line \\(m\\) passing through \\(F(0,1)\\) intersects the curve \\(C\\) at points \\(A\\) and \\(B\\). Tangents to curve \\(C\\) at points \\(A\\) and \\(B\\) are \\(l_{1}\\) and \\(l_{2}\\) respectively, and lines \\(l_{1}\\) and \\(l_{2}\\) intersect at point \\(M\\). Find the minimum value of the area of \\(\\triangle MAB\\).

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