
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 $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{1}{5}$ and for every positive integer $n$, 
\[ a_n = 3a_{n-1}^2 + \frac{1}{2}. \]
Let $d$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality 
\[ |a_0 a_1 \dotsm a_{n-1}| \le \frac{d}{3^n}. \]
What is the value of $100d$, rounded to the nearest integer?

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