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

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