Consider the following one-person game: A player starts with a score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it $a$) and write two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b \times c$ to her score. She repeats this step several times until she ends up with all $1$'s on the whiteboard. Then the game is over, and the final score is calculated. Let $M$ and $m$ be the maximum and minimum final scores that can be possibly obtained, respectively. Find $M - m$.

Let's think step by step and output the final answer within \boxed{}.
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