
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 the particular solution to the differential equation

$$
x^{4} \cdot y^{\prime \prime \prime}-2 x^{3} \cdot y^{\prime \prime}=\frac{5}{x}
$$

that satisfies the initial conditions \( y(1) = -\frac{1}{18} \), \( y^{\prime}(1) = -\frac{7}{18} \), and \( y^{\prime \prime}(1) = \frac{1}{6} \).

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