Okay, we are tasked with adding 5666123111 and 4356780669. Let's begin!

Step 1: We'll start by adding the digits 1 & 9 in column 1 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 2: We'll start by adding the digits 1 & 6 in column 2 and get 8.

Step 3: We'll start by adding the digits 1 & 6 in column 3 and get 7.

Step 4: We'll start by adding the digits 3 & 0 in column 4 and get 3.

Step 5: We'll start by adding the digits 2 & 8 in column 5 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 6: We'll start by adding the digits 1 & 7 in column 6 and get 9.

Step 7: We'll start by adding the digits 6 & 6 in column 7 and get 12.
We'll write down the last digit 2 and carry the 1 to the next column.

Step 8: We'll start by adding the digits 6 & 5 in column 8 and get 12.
We'll write down the last digit 2 and carry the 1 to the next column.

Step 9: We'll start by adding the digits 6 & 3 in column 9 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 10: We'll start by adding the digits 5 & 4 in column 10 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

So, 5666123111 + 4356780669 = 10022903780.

