
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
Define $u_0 = \frac{1}{3}$, and for $k \ge 0$ let $u_{k+1}$ follow the recurrence:
\[
u_{k+1} = 3u_k - 3u_k^2
\]
Determine the limit $L$ of this sequence, and calculate the smallest value of $k$ such that:
\[
|u_k - L| \leq \frac{1}{2^{1000}}
\]
A) 0
B) 1
C) 2
D) 10
E) 100

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