
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 king has traversed a chessboard, visiting each square exactly once and returning to the starting square with his final move. (The king moves according to standard chess rules: in one move, he can transition horizontally, vertically, or diagonally to any adjacent square.) When his path was drawn by consecutively connecting the centers of the squares he passed through, a closed broken line without self-intersections was formed. What is the minimum and maximum length that this line can have? (The side of each square is equal to one.)

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