
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 creative cultural company has developed a new toy (in units of sets) for production and sale. According to past experience, the cost $p$ (in units of yuan) of producing $x$ sets of toys per month consists of two parts:
a. Fixed costs (unrelated to the number of sets $x$ produced), totaling ten million yuan;
b. Direct total cost for production, $50x + \frac{1}{100}x^2$.
(1) What is the number of toy sets that the company should produce per month to minimize the average cost per set? What is the cost per set in this case?
(2) Assuming that all toys produced each month can be sold out, the direct total cost will increase sharply with the increase of $x$. Therefore, the selling price should also increase appropriately with the increase of $x$. Let $q$ be the selling price (in yuan) of each set of toys, where $q = a + \frac{x}{b}$ ($a$, $b \in \mathbb{R}$). If the profit is maximized when the production volume is 15,000 sets, and the price per set at this time is 300 yuan, find the values of $a$ and $b$. (Profit = sales revenue - production cost)

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