
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 $\overrightarrow{m}=(\sin x,1), \overrightarrow{n}=(\sqrt{3}\cos x,\frac{1}{2}\cos 2x)$, and the function $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}$.  
$(1)$ Find the maximum value of the function $f(x)$ and the corresponding set of independent variables.  
$(2)$ In triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, if $f(A)=\frac{1}{2}$ and $a=2$, find the maximum area of triangle $ABC$.

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