
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
Solve the equations:
1) \(\frac{d^{4} y}{d x^{4}}-3 \frac{d^{2} y}{d x^{2}}=9 x^{2}\)
2) \(\frac{d^{3} x}{d t^{3}}-3 \frac{d^{2} x}{d t^{2}}+2 \frac{d x}{d t}=4 e^{2 t}-3 e^{3 t}\)
3) \(4 y^{\prime \prime \prime}+y^{\prime}=3 e^{x}+2 \sin \frac{x}{2}\)
4) \(y^{\prime \prime \prime}+y^{\prime \prime}=1-6 x^{2} e^{-x}\)

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