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

(1) Find the smallest positive period and the monotonically decreasing interval of $f(x)$.

(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively. If $f(A)=4$, $b=1$, 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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