
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 points $A$, $B$, and $C$ as $A(3,0)$, $B(0,3)$, and $C(\cos \alpha ,\sin \alpha ), \alpha \in (\frac{\pi }{2},\frac{3\pi }{2})$,

(1) Find the value of angle $\alpha$ if $||\overrightarrow{AC} ||=||\overrightarrow{BC} ||$,

(2) Find the value of $\frac{2{\sin }^{2}\alpha+\sin 2\alpha}{1+\tan \alpha }$ if $\overrightarrow{AC} \cdot \overrightarrow{BC}=-1$.

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