
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 $\triangle ABC$ satisfies $4\sin B\sin \left(C-A\right)+1=4\sin A\sin B+\cos 2C$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively.  
$(1)$ Determine whether angle $C$ can be a right angle. Explain your reasoning.  
$(2)$ If $\triangle ABC$ is an acute triangle and satisfies $\sin B=\lambda \sin A$, find the range of real numbers for $\lambda$.

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