
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:

$C_{5}^{1} + C_{5}^{5} = 2^{3} - 2$,

$C_{9}^{1} + C_{9}^{5} + C_{9}^{9} = 2^{7} + 2^{3}$,

$C_{13}^{1} + C_{13}^{5} + C_{13}^{9} + C_{13}^{13} = 2^{11} - 2^{5}$,

$C_{17}^{1} + C_{17}^{5} + C_{17}^{9} + C_{17}^{13} + C_{17}^{17} = 2^{15} + 2^{7}$,

$({\cdots})$

From the above equations, infer a general conclusion: 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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