
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 $\alpha ,\beta \in \left[ 0,\frac{\pi}{2} \right]$, and $f\left( 4\alpha +\frac{4}{3}\pi \right)=-\frac{30}{17}$, $f\left( 4\beta -\frac{2}{3}\pi \right)=\frac{8}{5}$, find the value of $\cos (\alpha +\beta)$.

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