
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 the initial side of angle $\theta$ is the non-negative x-axis, and the terminal side passes through point $A(-1,\sqrt{2})$.  
$(1)$ Find the value of $\frac{{-cos({\frac{{3π}}{2}+θ})+\sqrt{2}sin({\frac{π}{2}+θ})}}{{sin(2π-θ)-2\sqrt{2}cos(-θ)}}$.  
$(2)$ Given that the initial side of angle $\alpha$ is the non-negative x-axis, and the terminal sides of angles $\theta$ and $\alpha$ are symmetric about the y-axis, find the value of $sin({α-\frac{π}{6}})$.

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