
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 vectors $\overrightarrow{a} = (1, \cos x)$ and $\overrightarrow{b} = \left(\frac{1}{3}, \sin x\right)$, where $x \in (0, \pi)$,
(I) If $\overrightarrow{a} \parallel \overrightarrow{b}$, find the values of $\tan x$ and $\frac{\sin x + \cos x}{\sin x - \cos x}$;
(II) 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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