
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 the $1005$ boxes in a line contains exactly one red marble. For $1 \le k \le 1005$, the box in the $k\text{th}$ position additionally contains $2k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, proceeding in order. She stops when she first draws a red marble. Define $P(n)$ as the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{1005}$?
A) 1003
B) 1004
C) 1005
D) 1006

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