
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 $\overrightarrow{a}=(\sin x,-\cos x)$, $\overrightarrow{b}=(\sqrt{3}\cos x,-\cos x)$, $f(x)=2\overrightarrow{a}\cdot\overrightarrow{b}$  
$(1)$ Find the analytical expression for $f(x)$;  
$(2)$ In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. If $f(A)=2$, $b=1$, and the area of $\triangle ABC$ is $\frac{\sqrt{3}}{2}$, find the value of $a$.

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