
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
Consider a sequence \(b_1, b_2, \ldots\) defined by the following properties:
- \(b_1 = 2\)
- \(b_{2n+1} = 2n \cdot b_n\) for any positive integer \(n\).

Calculate the value of \(b_{2^{100} + 1}\).

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