
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 finite geometric sequence \( S = (b, br, br^2, \ldots, br^9) \) consisting of 10 terms, where \( r > 0 \). Define \( B(S) \) to be the sequence 
\[ 
\left( \sqrt{b \cdot br}, \sqrt{br \cdot br^2}, \ldots, \sqrt{br^8 \cdot br^9} \right) 
\]
of 9 terms, obtained by taking the geometric mean of consecutive terms of \( S \). Define \( B^1(S) = B(S) \) and for each integer \( m \), \( 2 \leq m \leq 9 \), define \( B^m(S) = B(B^{m-1}(S)) \). If after 9 transformations \( B^9(S) = (r^4) \), find \( r \).
A) 0.5
B) 1
C) 1.5
D) 2

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