
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 $\overrightarrow{m}=( \sqrt {3}\sin \frac {x}{3},\cos \frac {x}{3})$ and $\overrightarrow{n}=(\cos \frac {x}{3},\cos \frac {x}{3})$, with $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$, find the smallest positive period and the symmetry center of function $f(x)$. Additionally, if $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $a=2$, $(2a-b)\cos C=c\cos B$, and $f(A)= \frac {3}{2}$, find $c$.

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