
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
Let the function $f(x) = m \sin x + \cos x$ ($x \in \mathbf{R}$) pass through the point $\left( \frac{\pi}{2}, 1 \right)$.

(Ⅰ) Determine the explicit formula for $y = f(x)$, and find the function's smallest positive period and extremum values.

(Ⅱ) If $f\left( \frac{\pi}{12} \right) = \sqrt{2}\sin A$, where $A$ is an angle of an acute triangle $\triangle ABC$ with an area of $\frac{3\sqrt{3}}{2}$ and $AB = 2$, find the lengths of $AC$ and $BC$.

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