
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 Maclaurin series for the functions \(e^{x}, \sin x, \cos x\), \((1+x)^{2}\), and \(\ln(1+x)\) and using the rules for multiplication and addition of power series, find the power series expansions for the following functions:

1. \((1+x) e^{x}\)
2. \(\sin^{2} x\)
3. \(\frac{x-3}{(x+1)^{2}}\)
4. \(e^{-x} \sin x\)
5. \(\ln\left(1+3x+2x^{2}\right)\)

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