
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  $a_0 = \frac{4}{5}$  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?

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