
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]


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A factory's total cost of producing a product consists of three parts: (1) material cost of 50 yuan per item; (2) labor costs of 7500 + 20x yuan; (3) power supply and machinery maintenance costs of $x^2 - 30x + 600$ yuan (where x is the number of products).  
(1) Express the cost per product P(x) in yuan as a function of the number of products x and determine the lowest cost per product.  
(2) If the product is in high demand, a market survey indicates that the selling price per product is $Q(x) = 1240 - \frac{1}{30}x^2$. What range of production volume will put the factory in a state of increasing productive capacity (meaning that if the production volume increases, the total profit will also increase)?

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