
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 in $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $\frac{\cos A-\cos C}{\cos B}=\frac{\sin C-\sin A}{\sin B}$.

$(1)$ Find the value of $\frac{c}{a}$;

$(2)$ If $\cos B=\frac{2}{3}$ and the area of $\triangle ABC$ is $\frac{\sqrt{5}}{6}$, find the value of $b$.

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