
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider the following one-person game: A player starts with score  $0$  and writes the number  $20$  on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes 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 the 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  $M, m$  be the maximum and minimum final score that can be possibly obtained respectively.
Find  $M-m$ .

Present the answer in LaTex format: \boxed{Your answer}
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