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

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