
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 vectors $\overset{→}{a}=(\cos x,-1)$ and $\overset{→}{b}=( \sqrt{3}\sin x,-\frac{1}{2})$, and the function $f(x)=( \overset{→}{a}+\overset{→}{b})\overset{→}{a}-2$.

(1) Find the smallest positive period and the interval where the function is monotonically increasing.
(2) In triangle $ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The graph of function $f(x)$ passes through point $(A,\frac{1}{2})$, and $b$, $a$, $c$ form an arithmetic sequence. Also, $\overset{→}{AB}\cdot\overset{→}{AC} = 9$. Find the value of $a$.

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