
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 $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence \[u_{k+1} = 3u_k - 3u_k^2.\]
This sequence approaches a limit; denote it as $L$. Determine the smallest value of $k$ such that \[|u_k-L| \le \frac{1}{2^{2000}}?\]
A) 10
B) 12
C) 14
D) 16
E) 18

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