
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 $\triangle ABC$, let $a$, $b$, $c$ be the lengths of the sides opposite to angles $A$, $B$, $C$ respectively, and $\left(2a-b\right)\cos C=c\cos B$.  
$(1)$ Find angle $C$;  
$(2)$ If $a=2$, $b=3$, $CD$ is the angle bisector of angle $C$, find the length of $CD$;  
$(3)$ If $a\cos B+b\cos A=4$, find the range of values for the area of acute triangle $\triangle ABC$.

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