
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 60 empty boxes \( B_{1}, B_{2}, \ldots, B_{60} \) in a row on a table and an unlimited supply of pebbles. Given a positive integer \( n \), Alice and Bob play the following game.

In the first round, Alice takes \( n \) pebbles and distributes them into the 60 boxes as she wishes. Each subsequent round consists of two steps:
(a) Bob chooses an integer \( k \) with \( 1 \leq k \leq 59 \) and splits the boxes into the two groups \( B_{1}, \ldots, B_{k} \) and \( B_{k+1}, \ldots, B_{60} \).
(b) Alice picks one of these two groups, adds one pebble to each box in that group, and removes one pebble from each box in the other group.

Bob wins if, at the end of any round, some box contains no pebbles. Find the smallest \( n \) such that Alice can prevent Bob from winning.

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