
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 acute triangle $\triangle ABC$, the sides opposite to $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{{{c^2}+{a^2}-{b^2}}}{{2c}}=2b\cos^2\frac{A}{2}$.  
$(1)$ Find the range of $\cos B$;  
$(2)$ Find the range of $\frac{{2a+c}}{b}$.

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