
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 a Fourier series in the specified interval:
1) \(\varphi(x)=\frac{x}{2} ; \quad(0,2 \pi)\)
2) \(\psi(x)=e^{-x} ;(-\pi, \pi)\)
3) \(y=\begin{cases}6 & \text{for } 0<x<2 \\ 3x & \text{for } 2<x<4 \end{cases}\)
4) \(u=|\sin x| ;[-\pi, \pi]\)

Using the obtained decomposition, find the sum \(S\) of the series \(\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots+\frac{1}{(2k-1)(2k+1)}+\ldots\)

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