
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 $1000$ boxes in a line contains a single red marble, and for $1 \leq k \leq 1000$, the box in the $k\text{ th}$ position also contains $2k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let \( P(n) \) be the probability that Isabella stops after drawing exactly \( n \) marbles. What is the smallest value of \( n \) for which \( P(n) < \frac{1}{1000} \)?
A) 30
B) 31
C) 32
D) 33
E) 34

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