
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 $f(x) = \sqrt{3}\sin x + \cos x + 1$.
(1) Find the smallest positive period and the range of the function $f(x)$.
(2) If $\alpha$ is an angle in the third quadrant, and $f(\alpha - \frac{\pi}{6}) = \frac{1}{3}$, find the value of $\frac{\cos 2\alpha}{1 + \cos 2\alpha - \sin 2\alpha}$.

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