
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 \( v_0 = -\frac{3}{8}\), and for \( n \geq 0 \), let \( v_{n+1} \) be determined by the recurrence \[ v_{n+1} = -v_n^2. \]
This sequence tends to a limit; call it \( M \). What is the least value of \( n \) such that \[ |v_n - M| \leq \frac{1}{2^{500}}? \]
A) 6
B) 7
C) 8
D) 9
E) 10

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