
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 $f(x) = 2\cos x\sin\left(x + \frac{\pi}{3}\right) - \sqrt{3}\sin^2 x + \sin x\cos x$,
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the minimum value of $f(x)$ and the corresponding values of $x$ when the minimum is achieved;
(3) If the inverse function of $f(x)$, denoted by $f^{-1}(x)$, exists when $x \in \left[\frac{\pi}{12}, \frac{7\pi}{12}\right]$, find the value of $f^{-1}(1)$.

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