
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
The following game is played with a group of  $n$  people and  $n+1$  hats are numbered from  $1$  to  $n+1.$  The people are blindfolded and each of them puts one of the  $n+1$  hats on his head (the remaining hat is hidden). Now, a line is formed with the  $n$  people, and their eyes are uncovered: each of them can see the numbers on the hats of the people standing in front of him. Now, starting from the last person (who can see all the other players) the players take turns to guess the number of the hat on their head, but no two players can guess the same number (each player hears all the guesses from the other players).

What is the highest number of guaranteed correct guesses, if the  $n$  people can discuss a common strategy?

*Proposed by Viktor Kiss, Budapest*

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