Consider how the perfect outer shuffle rearranges the positions of elements in an array and try to model this rearrangement mathematically.

Observe that the shuffle can be viewed as two interleaved sequences—one for the first half and one for the second half—and derive a formula that maps the original index to its new position based on these sequences.

Generalize the pattern by analyzing the binary representation of indices or their parity (odd/even) to derive a closed-form expression for the new position after the shuffle.