
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 $\overrightarrow{a}=(\sqrt{3}\sin x,\cos x)$ and $\overrightarrow{b}=(\sin(x+\frac{\pi}{2}),\cos x)$. Let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(1)$ Find the smallest positive period of the function $y=f(x)$.  
$(2)$ In triangle $ABC$, with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $f(A)=1$, $b=4$, the area of triangle $ABC$ is $2\sqrt{3}$, find the length of side $a$.

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