
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 $\frac{\cos x-\sin x}{\cos y}=2 \sqrt{2} \operatorname{ctg} \frac{x+y}{2}$ and $\frac{\sin x+\cos x}{\sin y}=\frac{1}{\sqrt{2}} \operatorname{tg} \frac{x+y}{2}$. Find all possible values of the expression $\operatorname{tg}(x+y)$, given that there are at least three of them.

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