
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
There are 2023 boxes arranged in a line, where each box contains a single blue marble. Additionally, for $1 \leq k \leq 2023$, the box in the $k$-th position contains $2k$ green marbles. Michael begins at the first box and draws a single marble at random from each box, proceeding in order. He stops when he first draws a blue marble. Define $P(n)$ as the probability that Michael stops after drawing exactly $n$ marbles. Determine the smallest value of $n$ for which $P(n) < \frac{1}{2023}$.
A) 30
B) 31
C) 32
D) 33
E) 34

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