
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 $\overrightarrow{m}=(\cos x,-1)$ and $\overrightarrow{n}=(\sqrt{3}\sin x,-\frac{1}{2})$, let the function $f(x)=(\overrightarrow{m}+\overrightarrow{n})\cdot\overrightarrow{m}$.

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

$(2)$ Given $a$, $b$, and $c$ are the lengths of the sides opposite to the angles of triangle $ABC$, respectively, with $A$ being an acute angle, $a=1$, $c=\sqrt{3}$, and $f(A)$ is exactly the maximum value of the function $f(x)$ on the interval $\left[0,\frac{\pi}{2}\right]$, find $A$, $b$, and the area of triangle $ABC$.

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