
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 $\overrightarrow{a}=(\frac{1}{2},\frac{1}{2}\sin x+\frac{\sqrt{3}}{2}\cos x)$, $\overrightarrow{b}=(1,y)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the function be $y=f(x)$.

$(1)$ Find the smallest positive period of the function $y=f(x)$;

$(2)$ Given an acute triangle $ABC$ with angles $A$, $B$, and $C$, if $f(A-\frac{\pi}{3})=\sqrt{3}$, side $BC= \sqrt{7}$, $\sin B=\frac{\sqrt{21}}{7}$, find the length of $AC$ and the area of $\triangle ABC$.

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