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Therefore, both intervals align after:
Thus, the total number of ways to assign flight patterns is calculated by multiplying the number of ways to choose the unused pattern by the number of ways to assign the remaining patterns:
To return to the original position after four days, the birds movements must cancel out in both the north-south and east-west directions. Consider north and south as one axis and east and west as another.
Also, the total number of movement days is 4:
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