Find the number of 10-digit sequences where the difference between any two consecutive digits is 1, using **only** the digits 1, 2, 3, and 4.

Examples of such 10-digit sequences are 1234321232 and 2121212121.

**Bonus:** Let \(T(n)\) be the number of such \(n\)-digit sequences. What is \( \lim_{n \to \infty} \frac{T(n+1)}{T(n)}?\)

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