
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
Determine the smallest natural number \( k \) such that for any \( a \in [0,1] \) and any \( n \in \mathbf{N}^{*} \), it always holds that \( a^{k} (1-a)^{n} < \frac{1}{(n+1)^{3}} \).

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