
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 $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(-\sqrt{6}, \sqrt{2})$, with $x \in [0,\pi]$.
1. If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $x$.
2. Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$, determine the maximum and minimum values of $f(x)$ and the corresponding values of $x$.

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