
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 $f(x)=2\cos ^{2}x+\sin ( \frac {7\pi}{6}-2x)-1(x\in R)$;
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes through the point $(A, \frac {1}{2})$. If $\overrightarrow{AB}^{2}- \overrightarrow{AC}\cdot \overrightarrow{CB}- \overrightarrow{BC}=4$, find the minimum value of $a$.

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