
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 $f\left( x \right)=2\cos^{2}x+\sin\left( \frac{7\pi }{6}-2x \right)-1\left( x\in R \right)$:

(1) Find the smallest positive period and the monotonically increasing interval of the function $f\left( x \right)$;

(2) In $\Delta ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f\left( x \right)$ passes through point $\left( A,\frac{1}{2} \right)$, and $b$, $a$, $c$ form an arithmetic sequence, with $\overrightarrow{CA}\cdot \overrightarrow{AB}=-9$. Find the value of $a$.

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