Chloe drove her new car on a trip for a whole number of hours, averaging $77$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a 3-digit number with $a\ge 1$ and $a+b+c\le 9$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2$?