
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 $\overrightarrow{a}=({\cos x, \sin x})$ and $\overrightarrow{b}=({2\cos x, 2\sqrt{3}\cos x})$.

$(1)$ If $x\in \left[0,\pi \right]$, find the value of $x$ when $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$.

$(2)$ Let $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$.

$($i$)$ Find the smallest positive period of $f(x)$.

$($ii$)$ When $x\in \left[0,m\right]$, $f(x)$ can achieve $2$ maximum values. Find the range of values for $m$.

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