
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
A king traversed a chessboard, visiting each square exactly once and returning to the starting square with the last move. (The king moves according to the usual rules: in one move he can move horizontally, vertically, or diagonally to any neighboring square.) When the king's path was drawn by sequentially connecting the centers of the squares he visited, the result was a closed broken line without self-intersections. What is the minimum and the maximum length it can have? (The side of each cell is one unit.)

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