
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
60 boxes are arranged in a line on a table. For a certain natural number $n > 0$, Alice and Bob play the following game. Alice starts by distributing $n$ marbles among the boxes as she wishes. Then, each turn proceeds in two steps:

(a) Bob chooses an integer $1 \leq k \leq 59$ and separates the boxes into two groups: $B_{1}, \ldots, B_{k}$ and $B_{k+1}, \ldots, B_{60}$.

(b) Alice chooses one of these two groups and adds one marble to each box in this group, then removes one marble from each box in the other group.

Bob wins if Alice can no longer play, that is, if she has to remove a marble from an empty box. Find the minimal $n$ such that Alice can prevent Bob from winning.

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