
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
In a pile you have 100 stones. A partition of the pile in  $ k$  piles is *good* if:

1) the small piles have different numbers of stones;

2) for any partition of one of the small piles in 2 smaller piles, among the  $ k \plus{} 1$  piles you get 2 with the same number of stones (any pile has at least 1 stone).

Find the maximum and minimal values of  $ k$  for which this is possible.

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