
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
Let  $n$  be a positive integer. There is an infinite number of cards, each one of them having a non-negative integer written on it, such that for each integer  $l \geq 0$ , there are exactly  $n$  cards that have the number  $l$  written on them. A move consists of picking  $100$  cards from the infinite set of cards and discarding them. Find the least possible value of  $n$  for which there is an infinitely long series of moves such that for each positive integer  $k$ , the sum of the numbers written on the  $100$  chosen cards during the  $k$ -th move is equal to  $k$ .


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