
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 $\overrightarrow{m}=( \sqrt {3}\sin 2x+2,\cos x)$, $\overrightarrow{n}=(1,2\cos x)$, and the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.  
$(1)$ Find the smallest positive period of the function $f(x)$ and its range on $(- \dfrac {\pi}{6}, \dfrac {\pi}{2}]$;  
$(2)$ In $\triangle ABC$, if $f(A)=4$, $b=4$, and the area of $\triangle ABC$ is $\sqrt {3}$, find the value of $a$.

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