
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
Reading and Understanding:  
Given the formulas: $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$; $\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$; we can transform the function $g(x) = \sin x + \sqrt{3}\cos x$ into: $g(x) = 2\left(\frac{1}{2}\sin x + \frac{\sqrt{3}}{2}\cos x\right) = 2\left(\sin x\cos \frac{\pi}{3} + \cos x\sin \frac{\pi}{3}\right) = 2\sin\left(x + \frac{\pi}{3}\right)$  
(1) Based on your understanding, transform the function $f(x) = \sin x + \cos\left(x - \frac{\pi}{6}\right)$ into the form $f(x) = A\sin(\omega x + \varphi)$.  
(2) Determine the smallest positive period, symmetry center, and monotonic increasing interval of the function $f(x)$ from the previous question.

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