
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
Given the domain restriction \( x \neq 1 \), and considering that the denominator in this domain is negative, the inequality simplifies to:

$$
\sqrt{x^{2}-2x+2} \geq -\sqrt{5} x
$$

For \( x \geq 0 \), this inequality holds true. For \( x < 0 \), we have:

$$
x^{2}-2x+2 \geq 5x^{2}
$$

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