
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, and the southern has four times fewer residents. Both regions are able to produce goods X in quantity $x$ and Y in quantity $y$. Due to climatic conditions, the production possibilities of the regions differ. The production possibilities curve for one representative of the northern region is given by the equation $y=13.5-9x$, and for the southern region by $-y=24-1.5 \cdot x^{2}$. Both goods are produced solely for export, and sales are only possible in sets. Each set consists of 1 unit of goods X and 9 units of goods Y, and the quantity of sets can be non-integer. The price of a set is 6000 monetary units. Each region maximizes the total income from the sale of sets. All the 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 their production efforts to create sets and share the earned money, distributing it evenly among the residents of the entire country. The northern region's administration rejected this proposal, considering that it might reduce the region's total income. However, after careful calculation, the northern region stated that it is willing to give its production resources if the southern region pays an amount equal to the previous total income of the northern region, plus a symbolic fee of 1983 monetary units. Determine how the Gini coefficient will change if the southern region agrees to the northern region's terms.

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