
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 $\overrightarrow{a}=(1,\cos \alpha)$ and $\overrightarrow{b}=(\frac{1}{3},\sin \alpha)$, where $\alpha \in (0,\pi)$

$(1)$ If $\overrightarrow{a} \bot \overrightarrow{b}$, find the value of $\sin 2\alpha$

$(2)$ If $\overrightarrow{a} // \overrightarrow{b}$, find the value of $\frac{\sin \alpha +\cos \alpha}{\sin \alpha -\cos \alpha}$;

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