
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:
$$
C_5^1 + C_5^5 = 6,
$$
$$
C_9^1 + C_9^5 + C_9^9 = 2^7 + 2^3,
$$
$$
C_{13}^1 + C_{13}^5 + C_{13}^9 + C_{13}^{13} = 2^{11} - 2^5,
$$
$$
C_{17}^1 + C_{17}^5 + C_{17}^9 + C_{17}^{13} + C_{17}^{17} = 2^{15} + 2^7,
$$
$$
…
$$
From the given equations, infer a general conclusion:
For $n \in \mathbb{N}^*$, find the expression for
$$
C_{4n+1}^1 + C_{4n+1}^5 + C_{4n+1}^9 + \ldots + C_{4n+1}^{4n+1}.
$$

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