
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 $\overrightarrow{a}=(2\sqrt{3}\sin x, \cos x), \overrightarrow{b}=(\cos x, 2\cos x), f(x)=2\overrightarrow{a}\cdot\overrightarrow{b}+2m-1$.  
$(1)$ Find the expression of $f(x)$ in terms of $x$.  
$(2)$ If $x\in[0, \frac{\pi}{2}]$ and the minimum value of $f(x)$ is $3$, find the value of $m$.  
$(3)$ If $f(x)\leqslant 6$ for all $x\in[0, \frac{\pi}{2}]$, find the range of values for $m$.

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