
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
Let \( N \) be a positive integer. A collection of \( 4 N^2 \) unit tiles with two segments drawn on them as shown is assembled into a \( 2N \times 2N \) board. Tiles can be rotated.

![Tile Image](https://cdn.mathpix.com/cropped/2024_04_17_74a51d5ecbff9af6115bg-3.jpg?height=140&width=146&top_left_y=564&top_left_x=955)

The segments on the tiles define paths on the board. Determine the least possible number and the largest possible number of such paths.

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