
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 a function $f(x) = A \sin(\omega x + \varphi)$, where $x \in \mathbb{R}$, $A > 0$, $\omega > 0$, and $0 < \varphi < \frac{\pi}{2}$, the period of the function is $\pi$, and there is a lowest point on its graph at $M\left(\frac{2\pi}{3}, -2\right)$.  
(Ⅰ) Find the analytic expression of $f(x)$.  
(Ⅱ) Determine the intervals on which $f(x)$ is monotonic.  
(Ⅲ) Find the range of $f(x)$ when $x \in \left[0, \frac{\pi}{12}\right]$.

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