Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand. I got this question from a last year's olympiad paper.

Compute $999,999\cdot 222,222 + 333,333\cdot 333,334$.

Is there an approach to this by using pen-and-paper?
EDIT Working through on paper made me figure out the answer. Posted below.
I'd now like to see other methods. Thank you.

An algebra free way: the expression is
$$(10^6 -1)\bigg({2 \over 9}(10^6 - 1)\bigg) + {10^6 - 1 \over 3}{10^6 + 2 \over 3}$$
$$= {10^6 - 1 \over 3}\bigg({2 \over 3}(10^6 - 1) + {10^6 + 2 \over 3}\bigg)$$
$$= \bigg({10^6 - 1 \over 3}\bigg)10^6$$
$$= 333,333*1,000,000$$
$$= 333,333,000,000$$
