
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
Represent the left side of the equation as:

$$
\begin{gathered}
x^{4} + 4x - 1 = \left(x^{2}\right)^{2} + 2x^{2} - 2x^{2} + 1 - 1 + 4x - 1 = \\
=\left(x^{2} + 1\right)^{2} - \left(2x^{2} - 4x + 2\right) = \left(x^{2} + 1\right)^{2} - \left(\sqrt{2}(x - 1)\right)^{2} = \\
=\left(x^{2} + 1 - \sqrt{2}x + \sqrt{2}\right)\left(x^{2} + 1 + \sqrt{2}x - \sqrt{2}\right)
\end{gathered}
$$

Then, the original equation will be equivalent to the combination of two equations:

1) $x^{2} - \sqrt{2}x + 1 + \sqrt{2} = 0$, which has no roots as its discriminant is negative;
2) $x^{2} + \sqrt{2}x + 1 - \sqrt{2} = 0$, which has roots $x_{1,2} = \frac{-1 \pm \sqrt{2 \sqrt{2} - 1}}{\sqrt{2}}$.

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