
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 $\sin(x+y)=\frac{1}{3}$ and $\tan x=4\tan y$.

$(1)$ Find the value of $\sin\left(x-\frac{\pi}{6}\right)\sin\left(y-\frac{\pi}{3}\right)-\cos\left(x-\frac{\pi}{6}\right)\cos\left(y-\frac{\pi}{3}\right)$.

$(2)$ Find the value of $\sin(x-y)$.

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