
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
Let sequence \( b_1, b_2, \ldots \) be defined by the following properties:

(i) \( b_1 = 2 \),

(ii) \( b_{2n} = n \cdot b_n \) for any positive integer \( n \),

(iii) \( b_{2n+1} = b_{2n} + 1 \) for any positive integer \( n \).

What is the value of \( b_{2^{100}} \)?

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