
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 $b_0 = \frac{3}{5}$ 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