Randy is playing a simple game called "Guess the Number": there is a hidden integer $V$ whose value is between $1$ and $N$. The goal of the game is to guess this number $V$. In each round, Randy picks an integer between $1$ and $N$ as his guess. If the guess matches $V$, the game ends and he wins a prize. Otherwise, the game continues and asks Randy for his next guess, and so on.
In each round, Randy will use the following strategy to make his guess:
- He randomly selects an integer $x$ between $1$ and $N$.
- If $x$ was not tried in any previous round, $x$ would be his guess in the current round. This guess costs him \$1.
- Otherwise, as the guess in the current round, he would pick the smallest not tried integer that is larger than $x$ (but not larger than $N$).
This guess also costs him \$1.
- If no such number exists, he discards $x$ and goes to step $1$. Discarding number $x$ has no cost for Randy.
Question: what is the average amount of money Randy has to pay before he wins the prize?