
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 vectors $\overrightarrow{m}=(2\sin(x-\frac{\pi}{6}),1)$ and $\overrightarrow{n}=(2\cos x,1)$.  
$(1)$ If $\overrightarrow{m} \parallel \overrightarrow{n}$ and $x \in [0,2\pi]$, find the value of $x$.  
$(2)$ Let $f(x)=\overrightarrow{m} \cdot \overrightarrow{n}$. Find the maximum and minimum values of the function $f(x)$ on the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$.

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