
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 points $A$, $B$, $C$ with coordinates $A(3,0)$, $B(0,3)$, $C(\cos \alpha,\sin \alpha)$, where $\alpha\in\left( \frac {\pi}{2}, \frac {3\pi}{2}\right)$.  
$(1)$ If $| \overrightarrow{AC}|=| \overrightarrow{BC}|$, find the value of $\alpha$;  
$(2)$ If $ \overrightarrow{AC}\cdot \overrightarrow{BC}=-1$, 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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