
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
Let $u_0 = \frac{1}{3}$, and for $k \ge 0$, let $u_{k+1}$ be determined by the recurrence relationship \[u_{k+1} = 3u_k - 3u_k^3.\] The sequence converges to a limit; denote this limit as $L$. What is the smallest value of $k$ such that \[|u_k-L| \le \frac{1}{3^{300}}?\]
A) 1
B) 10
C) 87
D) 123
E) 329

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