Start this way:

7^0 = 001

7^1 = 007

7^2 = 049

7^3 = x43

7^4 = x01

7^5 = x07

7^6 = x49

7^7 = x43

7^8 = x01

7^9 = x07

Next to the last digit is either 0 or 4.

Notice that the digits occur in pairs.

The zero first occurs when the exponent is:

0,4,8,12,16,20,etc...

or in steps of 4.

900413 divided by 4 is some number with remainder of 1

Thus the last two digits of 7^900412 = 01

You need to do one more calculation.

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