
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
It is known that \(\frac{\cos x-\sin x}{\cos y}=\frac{\sqrt{2}}{3} \tan \left( \frac{x+y}{2} \right)\) and \(\frac{\sin x+\cos x}{\sin y}=3 \sqrt{2} \cot \left( \frac{x+y}{2} \right)\). Find all possible values of the expression \(\tan(x+y)\), given that there are at least three.

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