Alex decides to spend a year ($365$ days) in Albertland, a country with some very unusual fiscal policies. He pays his taxes in the form of one coin each day, with coins in Albertland having values of $1$, $2$, $3$, $4$, or $5$ alberts.
The government asks for a certain denomination of coin each day. However, once Alex pays a coin of value $n$, he is exempt from paying another coin of the same value for the next $n$ days, even if the government asks for one. How many distinct totals could Alex pay for that year?