Positively Described Numbers - Part 1

Two numbers are "amicable" if the sum of the proper factors of one number is equal to the other number. The first pair of "amicable" numbers is $$220$$ and $$284.$$ What is the sum of the numbers that make the second pair of "amicable" numbers?

$$\textbf{Details and Assumptions}$$

The proper factors of a number are the positive factors of the number that are not equal to the number itself. For example, the proper factors of $$12$$ are $$1,$$ $$2,$$ $$3,$$ $$4,$$ and $$6.$$

The proper factors of $$220$$ are $$1,$$ $$2,$$ $$4,$$ $$5,$$ $$10,$$ $$11,$$ $$20,$$ $$22,$$ $$44,$$ $$55,$$ and $$110.$$ The factors of $$284$$ are $$1,$$ $$2,$$ $$4,$$ $$71,$$ and $$142.$$ The numbers $$220$$ and $$284$$ are "amicable" because $$1+2+4+5+10+11+20+22+44+55+110=284$$ and $$1+2+4+71+142=220.$$

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