
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 the coordinates of \\(A\\), \\(B\\), and \\(C\\) are respectively \\(A(4,0)\\), \\(B(0,4)\\), and \\(C(3\cos \alpha,3\sin \alpha)\\).
\\((1)\\) If \\(\alpha \in (-\pi,0)\\) and \\(|\overrightarrow{AC}|=|\overrightarrow{BC}|\\), find the value of angle \\(\alpha\\);
\\((2)\\) If \\(\overrightarrow{AC} \perp \overrightarrow{BC}\\), find the value of \\(\frac{2\sin^2\alpha+\sin 2\alpha}{1+\tan \alpha}\\).

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