
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 circle \\(x^{2}+y^{2}-2ax-6ay+10a^{2}-4a=0(0 < a \leqslant 4)\\) with center \\(C\\), and a line \\(L\\): \\(y=x+m\\).

\\((1)\\) If \\(a=2\\), find the maximum length of the chord \\(|AB|\\) intercepted by circle \\(C\\) on line \\(L\\);

\\((2)\\) If \\(m=2\\), find the maximum length of the chord \\(|AB|\\) intercepted by circle \\(C\\) on line \\(L\\);

\\((3)\\) If line \\(L\\) is a tangent below the center \\(C\\) of the circle, as \\(a\\) varies, find the range of the real number \\(m\\).

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