
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
A chemical factory plans to produce a batch of chemical reagents soon. After conducting market research, it is known that the production cost of these reagents consists of three aspects:
1. The cost of raw materials for producing 1 unit of reagent is 50 yuan.
2. The total wages paid to all employees consist of a basic wage of 7500 yuan and a subsidy of 20 yuan for each unit of reagent produced.
3. The subsequent maintenance cost averages (x + 600/x - 30) yuan per unit (the total production volume is x units, and \(50 \leq x \leq 200\)).

(I) Express the cost of producing each unit of reagent as a function of x, \(P(x)\), and find the minimum value of \(P(x)\).
(II) If all products are sold, the sales revenue function \(Q(x)\) (in yuan) in relation to the production volume x (in units) is given by \(Q(x) = 1240x - \frac{1}{30}x^3\). What should the production volume be to maximize the profit for this batch of reagents?

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