Let T(n) be the number of tours over a 4 n playing board such that:

• The tour starts in the top left corner.
• The tour consists of moves that are up, down, left, or right one square.
• The tour visits each square exactly once.
• The tour ends in the bottom left corner.

The diagram shows one tour over a 4 10 board:

T(10) is 2329. What is T(10¹²) modulo 10⁸?