
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 consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. 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 elves or dwarves?

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