
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 $\overrightarrow {a} = (\cos x, -1)$ and $\overrightarrow {b} = (\sqrt{3}\sin x, -\frac{1}{2})$, consider the function $f(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{a} - 2$.
1. Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$;
2. In triangle $\triangle ABC$, with the opposite sides of angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, it is known that the graph of function $f(x)$ passes through the point $(A, \frac{1}{2})$, and $b$, $a$, $c$ form an arithmetic progression. Additionally, $\overrightarrow{AB} \cdot \overrightarrow{AC} = 9$. Find the value of $a$.

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