
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 $\overrightarrow{m}=(\sin\frac{x}{3}, -1)$, $\overrightarrow{n}=(\frac{\sqrt{3}}{2}A, \frac{1}{2}A\cos\frac{x}{3})$ ($A > 0$), the maximum value of the function $f(x)=\overrightarrow{n}\cdot\overrightarrow{m}$ is $2$.  
$(1)$ Find the smallest positive period and the analytical expression of $f(x)$.  
$(2)$ Let $\alpha, \beta \in [0, \frac{\pi}{2}]$, $f(3\alpha+ \frac{\pi}{2})= \frac{10}{13}$, $f(3\beta+2\pi)= \frac{6}{5}$, find the value of $\sin(\alpha-\beta)$.

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