Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and  
\[  
a_{n} = 2 a_{n-1}^2 - 1  
\]  
for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality  
\[  
a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.  
\]  
What is the value of $100c$, rounded to the nearest integer?

Let's think step by step and output the final answer within \boxed{}.
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