
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 fundamental set of solutions $y_{1}=\ln x, y_{2}=x$ of the corresponding homogeneous equation, find a particular solution to the equation

$$
x^{2}(1-\ln x) y^{\prime \prime}+x y^{\prime}-y=\frac{(1-\ln x)^{2}}{x}
$$

that satisfies the condition $\lim _{x \rightarrow+\infty} y=0$.

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