
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 $\overrightarrow{a}=(\cos x,\sin x)$, $\overrightarrow{b}=(\cos x+2 \sqrt{3},\sin x)$, $\overrightarrow{c}=(\sin a,\cos a)$, where $x\in\mathbb{R}$.

$(1)$ If $\overrightarrow{a} \perp \overrightarrow{c}$, find the value of $\cos (2x+2a)$;

$(2)$ If $a=0$, 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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