You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
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?
user
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rule
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?
train