Tours on a $4 \times N$ Playing Board
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 $10^8$?