
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 $f(x) = \sin(\frac{7\pi}{6} - 2x) - 2\sin^2 x + 1 (x \in \mathbb{R})$,
(1) Find the period of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes through points $(A, \frac{1}{2})$, $b$, $a$, $c$ forming an arithmetic sequence, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 9$. Find the value of $a$.

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