
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
Given that in triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a\sin A - b\sin B = 2\sin \left(A-B\right)$, and $a \neq b$. Find:
- $(1)$ the value of $c$;
- $(2)$ the range of values for $\omega$ such that the function $y = f\left(x\right)$, obtained by shifting the graph of $y = \sin x$ to the right by $\frac{\pi}{4}$ units and then upward by $c$ units, has exactly two extreme points in the interval $x \in (0, \pi)$.

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