
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 $u_0 = \frac{1}{3}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence \[u_{k+1} = 3u_k - 3u_k^2.\]
This sequence tends to a limit; call it $M$. What is the least value of $k$ such that \[|u_k - M| \le \frac{1}{2^{500}}?\]
A) 7
B) 8
C) 9
D) 10
E) 11

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