
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 $\overrightarrow{a}=(\cos ax,\sin ax)$ and $\overrightarrow{b}=(\sqrt{3}\cos ax,-\cos ax)$, where $a > 0$, if the graph of the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is tangent to the line $y=m(m > 0)$, and the abscissa of the tangent point forms an arithmetic sequence with a common difference of $\pi$.

$(1)$ Find the values of $a$ and $m$;  
$(2)$ In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. If $f\left( \frac{A}{2}\right) = \frac{\sqrt{3}}{2}$ and $a=4$, find the maximum area of $\triangle ABC$ and the values of $b$ and $c$ at this maximum.

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