
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
It is known that the pole \\(O\\) of the polar coordinate system coincides with the origin of the Cartesian coordinate system, and the polar axis coincides with the positive half-axis of the \\(x\\)-axis in the Cartesian coordinate system. The parametric equation of circle \\(C\\) is \\(\begin{cases}x=\alpha+\alpha\cos \theta \\\\ y=\alpha\sin \theta\end{cases} \\) (where \\(\theta\\) is the parameter, \\(0 < a < 5)\\), and the polar equation of line \\(l\\) is \\(\rho\sin (\theta+ \dfrac{\pi}{4}) =2\sqrt{2}\\). If line \\(l\\) intersects circle \\(C\\) at points \\(A\\) and \\(B\\), and \\(|AB|=2\sqrt{2}\\),

\\((1)\\) find the value of \\(a\\);  

\\((2)\\) If \\(M\\) and \\(N\\) are two points on circle \\(C\\) that do not coincide with point \\(O\\), and \\(\angle MON= \dfrac{\pi }{3}\\), find the maximum value of \\(|OM|+|ON|\\).

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