
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 into a Fourier series the function with period $2\pi$ given as follows:

$$
f(x)=\left[\begin{array}{c}
-1, \quad -\pi \leqslant x< 0 \\
\frac{1}{2}, \quad 0 < x \leqslant \pi 
\end{array}\right.
$$

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