
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
In a football league, whenever a player is transferred from a team  $X$  with  $x$  players to a team  $Y$  with  $y$  players, the federation is paid  $y-x$  billions liras by  $Y$  if  $y \geq x$ , while the federation pays  $x-y$  billions liras to  $X$  if  $x > y$ . A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with  $18$  teams of  $20$  players each. At the end of the season,  $12$  of the teams turn out to have again  $20$  players, while the remaining  $6$  teams end up with  $16,16, 21, 22, 22, 23$  players, respectively. What is the maximal amount the federation may have won during the season?

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