
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) = $\overrightarrow {a}$$$\cdot \overrightarrow {b}$$, where $\overrightarrow {a}$$ = (2\sin x, 1)$ and $\overrightarrow {b}$$ = (\sin(x+\frac{\pi}{3}), -\frac{1}{2})$,
1. Find the smallest positive period and symmetry center of f(x).
2. In triangle ABC, a, b, and c are the sides opposite to angles A, B, and C, respectively. If f(B) = 1, b = 2, and $\frac{b}{a}$$ = $\frac{\cos B+1}{2-\cos A}$$, find the area of triangle ABC.

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