
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 triangle \\(∆ABC\\), the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) are \\(a\\), \\(b\\), and \\(c\\) respectively. It is known that \\(c\tan C= \sqrt{3}(a\cos B+b\cos A)\\).

\\((1)\\) Find angle \\(C\\);

\\((2)\\) If point \\(D\\) is on side \\(BC\\), and \\(AD=CD=4\\), and the area of \\(∆ABD\\) is \\(8 \sqrt{3}\\), find the length of side \\(c\\).

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