
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
Beto plays the following game with his computer: initially the computer randomly picks  $30$  integers from  $1$  to  $2015$ , and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer  $k$  and some if the numbers written on the chalkboard, and subtracts  $k$  from each of the chosen numbers, with the condition that the resulting numbers remain non-negative. The objective of the game is to reduce all  $30$  numbers to  $0$ , in which case the game ends. Find the minimal number  $n$  such that, regardless of which numbers the computer chooses, Beto can end the game in at most  $n$  turns.

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