
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
Using the general formula, find the particular integral of the following differential equations:
1) \( y'' + 4y' + 4y = e^{-9x} \sec^2 x \)
2) \( y'' + 5y' + 6y = \left(e^{2x} + 1\right)^{-\frac{3}{2}} \)
3) \( y''' - 3y'' + 3y' - y = 5x^3 e^x + 3e^{2x} \)
4) \( y'' + 4y = \cos^3 x \)

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