
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 $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, $a$, $b$, and $c$ are the three sides, $a=2$, and $\cos C=-\frac{1}{4}$.  
$(1)$ If $\sin A=2\sin B$, find $b$ and $c$;  
$(2)$ If $\cos (A-\frac{π}{4})=\frac{4}{5}$, find $c$.

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