
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
A classroom contains $2023$ boxes in a line, each with a single blue marble. For boxes $1$ through $2023$, each box $k$ also contains exactly $k+1$ yellow marbles. Lila starts from the first box, drawing a single marble at random from each box progressively. She stops drawing when a blue marble is drawn for the first time, but she must draw at least one marble from each of the first $50$ boxes. Define $Q(n)$ as the probability that Lila stops drawing exactly at the $n$-th box. What is the smallest value of $n$ for which $Q(n) < \frac{1}{2023}$?
A) 1000
B) 1011
C) 1020
D) 1050
E) 1100

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