
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
An illusionist and his assistant are about to perform the following magic trick.

Let  $k$  be a positive integer. A spectator is given  $n=k!+k-1$  balls numbered  $1,2,…,n$ . Unseen by the illusionist, the spectator arranges the balls into a sequence as he sees fit. The assistant studies the sequence, chooses some block of  $k$  consecutive balls, and covers them under her scarf. Then the illusionist looks at the newly obscured sequence and guesses the precise order of the  $k$  balls he does not see.

Devise a strategy for the illusionist and the assistant to follow so that the trick always works.

(The strategy needs to be constructed explicitly. For instance, it should be possible to implement the strategy, as described by the solver, in the form of a computer program that takes  $k$  and the obscured sequence as input and then runs in time polynomial in  $n$ . A mere proof that an appropriate strategy exists does not qualify as a complete solution.)

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