
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
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\frac{{\cos(A+C)+\cos(A-C)-1}}{{\cos(A-B)+\cos C}}=\frac{c}{b}$.  
$(1)$ Find angle $B$;  
$(2)$ If $c=2$, point $D$ lies on side $AC$ such that $AD=2DC$, $BD=\frac{{2\sqrt{13}}}{3}$, find $b$.

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