
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
Set $v_0 = -\frac{1}{8}$, and for each $k \ge 0$ define the sequence,
\[ v_{k+1} = \frac{1}{2} v_k (1 - v_k). \]
This sequence approaches a limit; denote it by \( M \). Determine the smallest value of \( k \) such that 
\[ |v_k - M| \leq \frac{1}{2^{500}}. \]
A) 125  
B) 256  
C) 497  
D) 1000  
E) 2048

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