Let $T(n)$ be the number of tours over a $4 \times n$ 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 $4 \times 10$ board:

$T(10)$ is $2329$. What is $T(10^{12})$ modulo $10^8$?