
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 kingdom, the workforce is composed exclusively of the dwarf clan and the elf clan. Historically in this kingdom, dwarves and elves have always worked separately, and no enterprise ever hired both dwarves and elves simultaneously. The aggregate supply of labor resources for dwarves is represented by the function \( w_{\text{dwar}}^S = 1 + \frac{L}{3} \), and the aggregate supply of labor resources for elves is \( w_{\text{elf}}^S = 3 + L \). Conversely, the inverse function of aggregate demand for dwarf labor is \( w_{\text{dwar}}^D = 10 - \frac{2L}{3} \), and the inverse function of aggregate demand for elf labor is \( w_{\text{elf}}^D = 18 - 2L \).

The newly ascended king is very concerned about the difference in wage rates for his subjects and therefore decrees that the wages of elves and dwarves must be the same, ensuring that workers are not discriminated against based on their clan affiliation. The king believes that wage regulation may negatively impact the kingdom's economy and mandates that all his subjects behave completely competitively. By how many times will the wage of the group of workers, whose wage was lower before the king's intervention, increase if the firms in the kingdom are indifferent to hiring either elves or dwarves?

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