
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
Find all pairs of positive numbers \(x, y\) that satisfy the equation

$$
\begin{aligned}
& \frac{4 x^{2} y+6 x^{2}+2 x y-4 x}{3 x-y-2}+\sin \left(\frac{3 x^{2}+x y+x-y-2}{3 x-y-2}\right)=2 x y+y^{2}+\frac{x^{2}}{y^{2}}+\frac{2 x}{y}+ \\
& \quad+\frac{2 x y\left(x^{2}+y^{2}\right)}{(3 x-y-2)^{2}}+\frac{1}{(x+y)^{2}}\left(x^{2} \sin \frac{(x+y)^{2}}{x}+y^{2} \sin \frac{(x+y)^{2}}{y^{2}}+2 x y \sin \frac{(x+y)^{2}}{3 x-y-2}\right)
\end{aligned}
$$

In the answer, write the sum of the values \(x^{2} + y^{2}\) for all obtained solution pairs \((x, y)\) rounded to two decimal places if necessary. If there are no solutions, write -1, and if there are infinitely many solutions, write -2.

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