
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
Peter and Basil play the following game on a horizontal table  $1\times{2019}$ . Initially Peter chooses  $n$  positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by  $s$  cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by  $s$  cells neither to the left, nor to the right, the coin stays in the current cell. Find the least  $n$  such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

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