Lattices and Farey Sequences

[Vietnamese MO 2013] Write down some numbers $a_1,a_2,\ldots, a_n$ from left to right on a line. Step 1, we write $a_1+a_2$ between $a_1,a_2$; $a_2+a_3$ between $a_2,a_3$, …, $a_{n-1}+a_n$ between $a_{n-1},a_n$, and then we have new sequence $$b=(a_1, a_1+a_2,a_2,a_2+a_3,a_3, \ldots, a_{n-1}, a_{n-1}+a_n, a_n).$$ Step 2, we do the same thing with sequence b to have the new sequence c again…. And so on. If we do 2013 steps, count the number of the number 2013 appear on the line if

a) $n=2$, $a_1=1, a_2=1000$

b) $n=1000$, $a_i=i, i=1,2\ldots, 1000$

The value of answer (a) + answer (b) is?