
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 $f(x)= \frac{ \sqrt{3}}{2}\sin x\cos x+ \frac{1+\cos 2x}{4}$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively. If $f(A)= \frac{1}{2}$, $b+c=3$. Find the minimum value of $a$.

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