
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 the function $f(x)=A\cos\left(\frac{x}{4}+\frac{\pi}{6}\right)$, where $x\in\mathbb{R}$, and $f\left(\frac{\pi}{3}\right)=\sqrt{2}$  
$(1)$ Find the value of $A$;  
$(2)$ Let $\alpha,\beta\in\left[0,\frac{\pi}{2}\right]$, $f\left(4\alpha+\frac{4\pi}{3}\right)=-\frac{30}{17}$, $f\left(4\beta-\frac{2\pi}{3}\right)=\frac{8}{5}$, find the value of $\cos(\alpha+\beta)$.

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