Paranoid Cows

Farmer John has $N$ cows ($1 \le N \le 100{,}000$) who are self-conscious about their milk output. A cow who does not produce much milk is likely to be teased by the other cows.

FJ has set a milking schedule in which each cow $i$ ($1 \le i \le N$) is assigned a time interval $[A_i, B_i]$ ($0 \le A_i < B_i \le 1{,}000{,}000{,}000$). Cow $i$ enters the barn at time $A_i$ to be milked and leaves at time $B_i$. The barn door is narrow and fits only one cow at a time, so FJ made sure that no two cows enter or leave at the same time.

If the interval of cow $i$ contains the interval of cow $j$, that is, $A_i < A_j < B_j < B_i$, the two intervals nest. Nesting is bad. Cow $i$ stays in the barn for the whole of cow $j$'s milking and can spy on cow $j$ to learn her milk output. Cows do not want others to know their output, so FJ wants the schedule to contain no nested intervals.

Help FJ find the largest index $K$ ($1 \le K \le N$) such that the intervals of cows $1$ through $K$ contain no nesting.