Private nonprofit four$-$year colleges charge, on average, $$26,773$ per year in tuition and fees. The standard deviation is $$6,713.$ Assume the distribution is normal. Let X be the cost for a randomly selected college. Round all answers to $4$ decimal places where possible.

a$.$ What is the distribution of X$?$ X ~ N$($

b$.$ Find the probability that a randomly selected Private nonprofit four$-$year college will cost less than $23,062$ per year.

c$.$ Find the $80$th percentile for this distribution. $

$($Round to the nearest dollar.$)$