
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 $f(x)=2\sqrt{3}\sin x\cos x+1-2\sin^2x$ $(x\in\mathbb{R})$.

$(1)$ Find the smallest positive period and the maximum value of the function $f(x)$.

$(2)$ If $\alpha\in\left(0, \frac{\pi}{3}\right)$ and $f(\alpha)=2$, find the value of $\alpha$.

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