
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
Ali has a pouch with $n$ coins, and Baba has an initially empty pouch with $n-1$ coins. Baba plays the following game: at the beginning, the coins in one pouch are distributed into two pouches, placing $a_{1}$ coins in one and $b_{1}$ coins in the other $(a_{1}, b_{1} > 0)$, and writes the product $a_{1} b_{1}$ on the board. Then, in each step $k$ $(k=2, 3, \ldots)$, Baba selects a pouch containing at least two coins, distributes the coins into two empty pouches, placing $a_{k}$ coins in one and $b_{k}$ coins in the other $(a_{k}, b_{k} > 0)$, and writes the product $a_{k} b_{k}$ on the board.

The game ends when there is exactly 1 coin in each pouch. At this point, Ali calculates the sum of the products $a_{k} b_{k}$ written on the board and gives Baba that amount of gold.

What is the maximum amount of gold Baba can receive?

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