
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 moving point \\(P\\) in the plane above the \\(x\\)-axis, the difference between the distance from point \\(P\\) to \\(F(0,1)\\) and its distance to the \\(y\\)-axis equals \\(1\\).
\\((1)\\) Find the equation of the trajectory \\(C\\) of the moving point \\(P\\);
\\((2)\\) Let \\(A\\) and \\(B\\) be two points on curve \\(C\\), and the sum of the \\(x\\)-coordinates of \\(A\\) and \\(B\\) is \\(4\\).
\\(①\\) Find the slope of line \\(AB\\);
\\(②\\) Suppose \\(M\\) is a point on curve \\(C\\), the tangent line to \\(C\\) at \\(M\\) is parallel to line \\(AB\\), and \\(AM\\) is perpendicular to \\(BM\\), find the equation of line \\(AB\\).

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