
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 $\overrightarrow {m}=( \sqrt {3}\sin 2x+2, \cos x)$, $\overrightarrow {n}=(1, 2\cos x)$, let the function $f(x)= \overrightarrow {m}\cdot \overrightarrow {n}$.  
(1) Find the smallest positive period of $f(x)$ and the interval where it is monotonically decreasing;  
(2) In $\triangle ABC$, where $a$, $b$, 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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