
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 $f(x)=\sin (2x-\frac{\pi }{6})+2\cos ^{2}x-1$ $(x\in\mathbb{R})$.

(Ⅰ) Find the intervals of monotonic increase for $f(x)$;

(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $f(A)=\frac{1}{2}$, and $b$, $a$, $c$ form an arithmetic sequence, and $\overrightarrow{AB}\cdot \overrightarrow{AC}=9$. Find the value of $a$.

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