
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 \( u_0 = \frac{1}{8} \), and for \( k \ge 0 \) let \( u_{k+1} \) be determined by the recurrence \[ u_{k+1} = 2u_k - 2u_k^2. \]
This sequence tends to a limit; call it \( L \). What is the least value of \( k \) such that \[ |u_k - L| \le \frac{1}{2^{10}}? \]
A) 2
B) 3
C) 4
D) 5
E) 6

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