
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:
$$
\begin{array}{l}
\mathrm{C}_{5}^{1}+\mathrm{C}_{5}^{5}=2^{3}-2, \\
\mathrm{C}_{9}^{1}+\mathrm{C}_{9}^{5}+\mathrm{C}_{9}^{9}=2^{7}+2^{3}, \\
\mathrm{C}_{13}^{1}+\mathrm{C}_{13}^{5}+\mathrm{C}_{13}^{9}+\mathrm{C}_{13}^{13}=2^{11}-2^{5}, \\
\mathrm{C}_{17}^{1}+\mathrm{C}_{17}^{5}+\mathrm{C}_{17}^{9}+\mathrm{C}_{17}^{13}+\mathrm{C}_{17}^{17}=2^{15}+2^{7}, \\
\ldots \ldots .
\end{array}
$$

Based on the above equations, derive a general conclusion: For
$$
n \in \mathbf{Z}_{+}, \sum_{i=0}^{n} \mathrm{C}_{4 n+1}^{4 i+1}=
$$

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