
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 \( v_0 = \frac{1}{3} \), and for each \( k \geq 0 \), define \( v_{k+1} \) by the recurrence relation:
\[
v_{k+1} = \frac{3}{2}v_k - \frac{3}{2}v_k^2.
\]
This sequence converges to a limit \( M \). Determine the smallest value of \( k \) such that 
\[
|v_k - M| \leq \frac{1}{2^{20}}.
\]
A) 3
B) 4
C) 5
D) 6

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