
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 $\overrightarrow{a}=(1,\cos x)$ and $\overrightarrow{b}=\left( \frac{1}{2},\sin x \right)$, where $x\in(0,\pi)$  
$(1)$ If $\overrightarrow{a}\parallel \overrightarrow{b}$, find the value of $\frac{\sin x+\cos x}{\sin x-\cos x}$;  
$(2)$ If $\overrightarrow{a}\perp \overrightarrow{b}$, find the value of $\sin x - \cos x$.

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