
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\\): \\( \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)\\) whose right focus is \\(F(1,0)\\), and the right vertex is \\(A\\), with \\(|AF|=1\\).

\\((1)\\) Find the standard equation of the ellipse \\(C\\).

\\((2)\\) If a moving line \\(l\\): \\(y=kx+m\\) intersects the ellipse \\(C\\) at exactly one point \\(P\\), and intersects the line \\(x=4\\) at point \\(Q\\), is there a point \\(M(t,0)\\) such that \\(\overrightarrow{MP} \cdot \overrightarrow{MQ} = 0\\) holds? If it exists, find the value of \\(t\\); if not, explain why.

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