
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

Observe the following equations:

\[ C_{5}^{1}+C_{5}^{5}=2^{3}-2 \]
\[ C_{9}^{1}+C_{9}^{5}+C_{9}^{9}=2^{7}+2^{3} \]
\[ C_{13}^{1}+C_{13}^{5}+C_{13}^{9}+C_{13}^{13}=2^{11}-2^{5} \]
\[ C_{17}^{1}+C_{17}^{5}+C_{17}^{9}+C_{17}^{13}+C_{17}^{17}=2^{15}+2^{7} \]
\[ \cdots \]

Determine the general conclusion from the above equations:

For \( n \in \mathbf{N}^{*} \),

\[ C_{4n+1}^{1} + C_{4n+1}^{5} + C_{4n+1}^{9} + \cdots + C_{4n+1}^{4n+1} = \]

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