
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
Decompose the given function into an incomplete Fourier series containing only cosines or only sines in the specified interval:

1) \(\varphi(x)=\left\{\begin{array}{r}0.3 \text{ for } 0<x<0.5 \\ -0.3 \text{ for } 0.5<x<1 \text{.}\end{array}\right.\)

2) \(y=x \cos x\) in the interval from 0 to \(\pi\). Using the obtained decomposition, find the sum of the series \(\sum_{k=0}^{+\infty} \frac{4k^{2}+1}{\left(4k^{2}-1\right)^{2}}\).

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