
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 $\overrightarrow{a}=(2\cos^2x,\sin x),\overrightarrow{b}=(\frac{1}{2},\sqrt{3}\cos x)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(1)$ Find the smallest positive period and the interval of monotonic decrease of $f(x)$;  
$(2)$ In $\triangle ABC$, $A+B=\frac{7}{12}\pi$, $f(A)=1$, $BC=2\sqrt{3}$, find the length of side $AC$.

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