
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
In the Cartesian coordinate system, curve \\(C_{1}\\) is given by \\(x^{2}+y^{2}=1\\). By stretching the x-coordinates and y-coordinates of all points on curve \\(C_{1}\\) by a factor of \\(\sqrt{2}\\) and \\(\sqrt{3}\\) respectively, we obtain curve \\(C_{2}\\).  
\\((1)\\) Write the parametric equations for curve \\(C_{2}\\).  
\\((2)\\) Find the point \\(P\\) on curve \\(C_{2}\\) such that the distance from \\(P\\) to the line \\(l\\): \\(x+y-4\sqrt{5}=0\\) is maximized, and determine the maximum distance.

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