
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 $2N \times 2N$ board is covered by non-overlapping $1 \times 2$ dominoes. A rooked knight (a chess piece) traversed the board, visiting each cell exactly once (each move of the rooked knight is to a cell adjacent by side). We shall call a move a longitudinal move if it is a transition from one cell of a domino to the other cell of the same domino. What is

a) the greatest;

b) the least possible number of longitudinal moves?

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