
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 opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\left(b+2a\right)\cos C+c\cos B=0$.  
$(1)$ Find $C$;  
$(2)$ If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2$, find the minimum value of $2a+b$.

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