
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. Daniel and Merlijn are playing a game. Daniel
has  $k$  sheets of paper lying next to each other on a table, where  $k$  is a
positive integer. On each of the sheets, he writes some of the numbers
from  $1$  up to  $n$  (he is allowed to write no number at all, or all numbers).
On the back of each of the sheets, he writes down the remaining numbers.
Once Daniel is ﬁnished, Merlijn can ﬂip some of the sheets of paper (he is
allowed to ﬂip no sheet at all, or all sheets). If Merlijn succeeds in making
all of the numbers from  $1$  up to n visible at least once, then he wins.
Determine the smallest  $k$  for which Merlijn can always win, regardless of
Daniel’s actions.

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