
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 $\overrightarrow{m}=(\sin A,\cos A)$, $\overrightarrow{n}=( \sqrt {3},-1)$, and $\overrightarrow{m}\cdot \overrightarrow{n}=1$, with $A$ being an acute angle.  
$(1)$ Find the size of angle $A$;  
$(2)$ Find the range of the function $f(x)=\cos 2x+4\cos A\sin x(x\in\mathbb{R})$.

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