
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 $\overrightarrow{m}=(\sin x,1)$, $\overrightarrow{n}=(\sqrt{3}A\cos x, \frac{A}{2}\cos 2x)$ with $(A > 0)$, the maximum value of the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n}$ is 6.  
(a) Determine the value of $A$, as well as the equation of the axis of symmetry and the coordinates of the center of symmetry for the graph of the function;  
(b) Translate the graph of the function $y=f(x)$ to the left by $\frac{\pi}{12}$ units, and then reduce the x-coordinates of the resulting points by half, while keeping the y-coordinates unchanged, to obtain the graph of function $y=g(x)$. Find 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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