
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
In a line, there are $2023$ boxes numbered from 1 to 2023. Each box from $1$ to $1012$ contains $2$ red marbles and $k$ white marbles, where $k$ is the box number. Each box from $1013$ to $2023$ contains $1$ red marble and $k$ white marbles. Two players, Isabella and Liam, play a game. Isabella starts at the first box and chooses a marble at random from each box. Whenever she draws a red marble, she passes the turn to Liam who removes one white marble from the next box before Isabella resumes drawing. Isabella’s game ends when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Identify the smallest value of $n$ for which $P(n) < \frac{1}{2023}$.
- **A)** 45
- **B)** 51
- **C)** 63
- **D)** 101
- **E)** 122

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