
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
Each of a sequence of 1000 boxes contains a single red marble, and for each \(1 \le k \le 1000\), the box in the \(k\)-th position contains \(k\) white marbles as well. A person starts at the first box and randomly draws one marble out of the box, then moves to the next one, continuing this pattern. The drawing stops when the first red marble is drawn. Define \(P(n)\) to be the probability that the drawing stops after exactly \(n\) draws. What is the smallest value of \(n\) for which \(P(n) < \frac{1}{1000}\)?
A) 31
B) 32
C) 33
D) 34

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