
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 $\overrightarrow{a}=(\cos x, \sin x)$, $\overrightarrow{b}=(\cos x + 2\sqrt{3}, \sin x)$, and $\overrightarrow{c}=(0, 1)$, where $x \in \mathbb{R}$:
1. If $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\cos 2x$.
2. If the function $f(x) = \overrightarrow{a} \cdot (\overrightarrow{b} - 2\overrightarrow{c})$, find the maximum value of $f(x)$ and the corresponding $x$ value.

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