Tours on a $4 \times N$ Playing Board
Problem Statement
Let $T(n)$ be the number of tours over a $4 \times n$ playing board such that:
- 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 $T(10^{9\times 2^N})$ modulo $10^8$?
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You need to submit in the format: "N:problem(N)", possibly with multiple values at once, separated by commas, with $N$ between $1$ and $100$.
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