I Find Blackboards Cool

I have an infinite number of \(1's\) written on a blackboard.

Jake chooses \(2\) of the integers \(p\) and \(q\) and replaces them with \(\dfrac{p+q}{4}={ k }_{ 1 }\). (he removes \(p\) and \(q\) and then writes \(\dfrac{p+q}{4}\))

Now he repeats the process with the number \({ k }_{ 1 }\) and another integer to achieve \({ k }_{ 2 }\), and repeats again with the number \({ k }_{ 2 }\) and another integer to achieve \({ k }_{ 3 }\)

\[{ k }_{ 2 }=\frac { 1+{ k }_{ 1 } }{ 4 } ,\quad { k }_{ 3 }=\frac { 1+{ k }_{ 2 } }{ 4 } \]

Since he is an immortal, he does this again and again until he is left with only a single number...

Given that this number can be expressed as \(\frac{a}{b}\), find \(a+b\)


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