
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 $f(x)=\sin x\cdot \cos (x- \frac {π}{6})+\cos ^{2}x- \frac {1}{2}$.
(1) Find the monotonically increasing interval and the symmetry center of the function.
(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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