
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 $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $c=\frac{\sqrt{5}}{2}b$.

$(1)$ If $C=2B$, find the value of $\cos B$; $(2)$ If $\overrightarrow{AB} \cdot \overrightarrow{AC}=\overrightarrow{CA} \cdot \overrightarrow{CB}$, find the value of $\cos \left(B+\frac{\pi }{4}\right)$.

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