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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