
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 $\overrightarrow{m}=(\sqrt{3}\sin 2x-1,\cos x), \overrightarrow{n}=(1,2\cos x)$, define function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$.

(1) Find the smallest positive period and the interval where the function is monotonically decreasing.

(2) In $\triangle ABC$, $a,b,c$ are the sides opposite to angles $A,B,C$ respectively. If $f(A)=1,b=1,\triangle ABC$'s area is $\frac{\sqrt{3}}{2}$, find the value of $a$.

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