
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 $|\vec{AC}| = |\vec{BC}|$, find the value of $\alpha$;

$(2)$ If $\vec{AC} \cdot \vec{BC} = -1$, 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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