
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
Given a sequence of $3020$ boxes, each box $k$ (where $1 \le k \le 3020$) contains one blue marble and $3k$ white marbles, find the smallest value of $n$ for which the probability that Theodora stops after drawing exactly $n$ marbles is less than $\frac{1}{3020}$.

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