
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\frac{\cos B - 2\cos A}{2a - b} = \frac{\cos C}{c}$.

$(1)$ Find the value of $\frac{a}{b}$;

$(2)$ If angle $A$ is obtuse and $c=3$, find the range of values for $b$.

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