
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
Expand the function

\[ 
f(x) = \left\{
\begin{array}{ll}
1 & \text{for } 0 \leq x \leq 24 \\
\frac{32 - x}{8} & \text{for } 24 < x \leq 32
\end{array}
\right.
\]

using the system \(\left\{\cos \frac{k \pi}{l} x\right\}\).

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