
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 \( B_{1}, B_{2}, \cdots, B_{60} \) arranged on a table, and there are enough pebbles. Given a positive integer \( n \), Alice and Bob play a game.

In the first round, Alice takes \( n \) pebbles and distributes these \( n \) pebbles among the 60 boxes as she wishes. Each subsequent round contains two steps:
1. Bob chooses an integer \( k \) (where \( 1 \leqslant k \leqslant 59 \)), dividing the 60 boxes into two groups: \( B_{1}, B_{2}, \cdots, B_{k} \) and \( B_{k+1}, B_{k+2}, \cdots, B_{60} \);
2. Alice chooses one of these two groups. In the chosen group, she places one additional pebble in each box, and in the other group, she removes one pebble from each box.

If, after any round, there is a box without any pebbles, Bob wins. Find the minimum positive integer \( n \) such that Alice can always prevent Bob from winning.

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