
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 triangle $\triangle ABC$ with internal angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$, where $\frac{{3(a-b)}}{c}=\frac{{3\sin C-2\sin B}}{{\sin A+\sin B}}$.  
$(1)$ Find $\cos A$;  
$(2)$ If the area of $\triangle ABC$ is $2\sqrt{2}$, $AD$ is the angle bisector of angle $A$ intersecting side $BC$ at point $D$, find the maximum length of segment $AD$.

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