
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 $f(x) = \sqrt{3}\sin 2x - 2\sin^2 x$,

(1) If point $P(1, \sqrt{3})$ is on the terminal side of angle $\alpha$, find the value of $f(\alpha)$.

(2) If $x \in \left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, find the range of $f(x)$.

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