
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 the function $f(x)= \sqrt{3} \sin 2x+\cos 2x-1(x\in R)$;
(1) Write out the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(B)=0$, $\overrightarrow{BA}\cdot \overrightarrow{BC}= \frac{3}{2}$, and $a+c=4$, find the value of $b$.

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