
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
By grouping the terms, we obtain: \((4 y^{2}+1)(x^{4}+2 x^{2}+2)=8|y|(x^{2}+1)\).

Obviously, there are no solutions when \(y=0\). Dividing both sides by \(2|y|(x^{2}+1)\):

$$
\left(2|y|+\frac{1}{2|y|}\right)\left(x^{2}+1+\frac{1}{x^{2}+1}\right)=4
$$

It is known (and can be easily shown) that if \(a>0\), then \(a+\frac{1}{a} \geq 2\), with equality only when \(a=1\). Therefore, the right-hand side equals 4 only if \(2|y|=1\) and \(x^{2}+1=1\).

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