
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 integral of the equation that satisfies the given initial conditions:
1) \( y'' + 4y' + 5y = 0 \quad \text{with} \quad y(0) = -3, \; y'(0) = 0 \).
2) \( y''' + 3y'' + 3y' + y = 0 \quad \text{with} \quad y(0) = -1, \; y'(0) = 2, \; y''(0) = 3 \).

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