
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 vectors $\vec{m} = (\sin x, 1)$ and $\vec{n} = (\sqrt{3}A\cos x, \frac{A}{2}\cos 2x)$ with $A > 0$, the maximum value of the function $f(x) = \vec{m} \cdot \vec{n}$ is $6$.

1. Find $A$;
2. Translate the graph of the function $y = f(x)$ to the left by $\frac{\pi}{12}$ units, then compress the $x$-coordinates of all points in the obtained graph by half (while keeping the $y$-coordinates unchanged) to get the graph of the function $y = g(x)$. Determine the range of $g(x)$ on the interval $\left[0, \frac{5\pi}{24}\right]$.

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