
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 $\overrightarrow{m}=( \sqrt {3}\sin x,\cos x)$, $\overrightarrow{n}=(\cos x,\cos x)$, $\overrightarrow{p}=(2 \sqrt {3},1)$.  
$(1)$ If $\overrightarrow{m}\parallel \overrightarrow{p}$, find the value of $\overrightarrow{m}\cdot \overrightarrow{n}$;     
$(2)$ If $x\in(0, \dfrac {\pi}{3}]$, find the range of the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.

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