
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
Suppose you live in a small town and are well-acquainted with all its residents. The townspeople, recognizing your economic literacy, propose that you open a staffing agency where you represent all employable residents to employers in the labor market. The townspeople will pay you 1% of their earned wages, with no other means to find employment. You have calculated that your costs for placing townspeople into jobs are given by \(TC(L)=\frac{1}{3200}L^{2}\), where \(L\) is the number of labor hours that the townspeople will work at enterprises. The townspeople trust you to the extent that they are willing to work for any wage you negotiate on their behalf. The town has only one major enterprise that produces a finished product using the technology \(q=16\sqrt{L}\) and sells it in a perfectly competitive market at a price of 1.

After consideration, you realize there are two possible negotiation schemes with the enterprise:
1) You propose a wage rate \(w\), and the enterprise decides the quantity of labor (in hours) it will use at that wage rate.
2) The enterprise proposes a wage rate \(w\), and you decide the volume of employment (in hours) you will offer to the townspeople.

Regardless of the negotiation scheme, all employable citizens work the same number of hours.

(a) Which scheme would you personally prefer, and what amount would your staffing agency earn under this scheme?

(b) Is it truly beneficial for the townspeople that you represent their interests? Justify your answer.

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