Strange Exponent Sequence

This is a strange sequence involving exponents. It is as follows: You pick any positive number. Take the last digit. This is what you'll be working with. Look at the chart below. If the number is...

1: 1 Always one!
2: 2, 4, 8, 6 Repeating 4 digit even sequence.
3: 3, 9, 7, 1 Repeating 4 digit odd sequence.
4: 4, 6 Only two numbers.
5: 5 Always 5!
6: 6 Always 6, surprisingly for me.
7: 7, 9, 3, 1 Repeating 4 digit odd sequence.
8: 8, 4, 2, 6 Repeating 4 digit even sequence.
9: 9, 1 Only two numbers.
10: 0 Always ends in 0.

11: 1
12: 2, 4, 8, 6
13: 3, 9, 7, 1
14: 4, 6
15: 5

Notice something strange? I do! For one, it repeats after 10. That's why this is so about the last number. Also, for every number 1-10, no matter how many digits the cycle was, the digits in the cycle were always even if the original number was even and vice versa.

Now, after you got your numbers in the sequence, you know what the last numbers will be! Whatever your number is, it is the first number in the sequence. The next number in the sequence occurs when you multiply your number by itself, or to the exponent of two. The last number should be the next number in the sequence. After that, multiply it by itself again. The last number now should be the 3rd number in the sequence. After your sequence is done, it restarts.

You might need an example. Here is one:

1. You pick 7. The sequence is 7, 9, 3, 1!
2. 7x7=49.The 9 is the last number, and is the number in the sequence!
3. Multiply by 7 again. You end up with 343. The 3 fits!
4. 343x7=2,401, which fits the 1!
5. After the sequence is done, it repeats. Next is 16,807, which is the first number in the sequence!