
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 $\triangle ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite angles $A$, $B$, and $C$, respectively, and $\left(a+b\right)\left(\sin A-\sin B\right)=\left(c-b\right)\sin C$.  
$(1)$ Find the measure of angle $A$.  
$(2)$ If $\sin B$ is a root of the equation ${x^2}-2x+\frac{{99}}{{100}}=0$, find the value of $\cos C$.

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