People from $80$ countries participated in the $1$st round of $IGMO$. In order to ensure that participants from any of the $80$ countries can travel to any one of the remaining $79$ countries by at most $2$ flights, the countries have an agreement such that among the $80$ countries, each country's airport connects to at least $n$ other countries' airport. $\bullet$ Find the minimum value of $n$, proving that it is the minimum. $\bullet$ Prove that for this minimum $n$, if there isn't a direct flight between $2$ countries, then there are at least $2$ paths that require $2$ flights between the countries Note: For two airports $A$ and $B$ to be connected, it must be possible to have a flight from $A$ to $B$ and a flight from $B$ to $A$. This adds $1$ to both the number of connections from the airports. Also note that for this problem, each country has only $1$ airport.