
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 vectors $\overrightarrow{m}=(\cos \alpha- \frac{\sqrt{2}}{3},-1)$ and $\overrightarrow{n}=(\sin \alpha,1)$, vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ are collinear, and $\alpha \in [-\frac{\pi}{2},0]$.  
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;  
$(2)$ Find the value of $\frac{\sin 2\alpha}{\sin \alpha - \cos \alpha}$.

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