Kay

solved in awk! wow!! wahoo!!

Part One:

NF { acc += $0 }
!NF {
    if (acc > record) { record = acc; }
    acc = 0;
}
END { print record }

Part Two (I switched from BSD awk to GNU awk because I'm laaazy!)

NF { sums[acc] += $0 }
!NF { acc++ }
END {
    asort(sums);
    print sums[length(sums)] + sums[length(sums)-1] + sums[length(sums)-2]
}