
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 \( a_0, a_1, a_2, \ldots \) be an infinite sequence of real numbers where \( a_0 = \frac{7}{17} \) and for every positive integer \( n \),
\[ a_n = 2a_{n-1}^2 - 1. \]
Let \( c \) be the smallest number such that for every positive integer \( n \), the product of the first \( n \) terms satisfies
\[ |a_0 a_1 \ldots a_{n-1}| \le \frac{c}{2^n}. \]
What is the value of \( 100c \), rounded to the nearest integer?

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