
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and satisfy $b^{2}+c^{2}-a^{2}=\frac{{4\sqrt{2}}}{3}bc$.  
$(1)$ Find the value of $\sin A$;  
$(2)$ If $\frac{{3c}}{a}=\frac{{\sqrt{2}\sin B}}{{\sin A}}$, and the area of $\triangle ABC$ is $2\sqrt{2}$,  
$($i) Find the value of side $c$;  
$($ii) Find the value of $\sin (2C-\frac{π}{6})$.

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