
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 $f(x) = a \sin x + b \cos x$ and $g(x) = 2\sin(\omega x + \frac{\pi}{3}) + 1$, if the function $f(x)$ and $g(x)$ have the same axis of symmetry, then the solution set of the inequality $g(x) > 2$ is ( ).

A) $(k\pi - \frac{\pi}{6}, k\pi + \frac{\pi}{2}) \quad (k\in \mathbb{Z})$

B) $(2k\pi - \frac{\pi}{6}, 2k\pi + \frac{\pi}{2}) \quad (k\in \mathbb{Z})$

C) $(2k\pi, 2k\pi + \frac{\pi}{6}) \quad (k\in \mathbb{Z})$

D) $(k\pi, k\pi + \frac{\pi}{6}) \quad (k\in \mathbb{Z})$

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