
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 \( b_0, b_1, b_2, \dots \) be an infinite sequence of real numbers such that \( b_0 = \frac{3}{7} \) and
\[
  b_{n} = 2 b_{n-1}^2 - 1
\]
for every positive integer \( n \). Let \( d \) be the smallest number such that for every positive integer \( n \), the product of the first \( n \) terms satisfies the inequality
\[
|b_0 b_1 \dotsm b_{n - 1}| \le \frac{d}{3^n}.
\]
What is the value of \( 100d \), rounded to the nearest integer?

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