
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 with non-overlapping $1 \times 2$ dominoes. A lame rook has moved across the board, visiting each cell exactly once (each move made by the lame rook is to a cell adjacent by side). We define a move as longitudinal if it transitions from one cell of a domino to the other cell of the same domino. Find

a) the maximum;

b) the minimum possible number of longitudinal moves.

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