
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 $\overrightarrow{a}=(\sin x, 1)$, $\overrightarrow{b}=(\sqrt{3}\cos x, -2)$, and function $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot\overrightarrow{a}$.  
$(1)$ If $\overrightarrow{a}\parallel\overrightarrow{b}$, find the value of $\cos 2x$;  
$(2)$ Given that $\triangle ABC$ is an acute triangle, $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in $\triangle ABC$, $b=2$, and $f(A)=\frac{1}{2}$, find the range of values for the area of $\triangle ABC$.

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