
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

A lawyer and an economist received an odd number of coins of the same value (more than four) for their joint work. They can divide the coins in only one of three ways, each of which is completed in three steps, with each pursuing their personal interest (each wants to get more coins):

Step 1: The lawyer divides all the coins into two arbitrary parts, each with at least two coins.

Step 2: The economist divides each part again into two parts, each with at least one coin.

The methods differ only in the third step:

Step 3:

Method 1: The economist takes the largest and smallest parts.

Method 2: The economist takes both middle parts.

Method 3: The economist chooses method 1 or 2 himself but gives one coin to the lawyer for the right to choose.

The economist won in the division. Which method did he choose and why?

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