
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 certain country, there are two regions - the northern and the southern. The northern region has 24 residents, while the southern has four times fewer residents. Both regions can produce goods X in quantity $x$ and goods Y in quantity $y$. The production capabilities of the regions differ due to climate peculiarities. The production possibility frontier for a representative of the northern region is given by the equation $y=13.5-9x$, and for the southern region by $-y=24-1.5x^2$. Both goods are produced solely for export, and sales are possible only in sets. Each set consists of 1 unit of goods X and 9 units of goods Y, and the number of sets can be fractional. The price of a set is 2000 monetary units. Each region maximizes the total income from selling the sets. All income earned by the region is evenly distributed among the residents of that region.

(a) Find the Gini coefficient in the country if both regions operate separately from each other.

(b) The southern region proposed to the northern region to combine production efforts in creating the sets and to split the earned money evenly among all the residents of the country. The northern region's administration refused this proposal, arguing that it might reduce the region's total income. However, after careful calculations, it stated that the northern region is ready to provide its production resources if the southern region pays an amount equal to the previously combined income of the northern region plus a symbolic fee of 661 monetary units. Determine how the Gini coefficient will change if the southern region agrees to the conditions of the northern region.

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