
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 $\overrightarrow{a}=(\cos x, \sin x)$, $\overrightarrow{b}=(\cos x+2\sqrt{3}, \sin x)$, $\overrightarrow{c}=(0,1)$, where $x\in \mathbb{R}$. <br/>$(1)$ If $\overrightarrow{a} \cdot \overrightarrow{c}=0$, find the value of $\cos 2x$; <br/>$(2)$ Let $f(x)=\sin\left(x+\frac{\pi}{3}\right)+\sin\left(x-\frac{\pi}{6}\right)$. Find the maximum value of the function $f(x)$ and determine the corresponding $x$ value.

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