
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 factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P(x)$ represent the daily profit and average profit (average profit = $\frac{\text{total profit}}{\text{total quantity}}$) for producing $x$ units of product, respectively.  
(1) At what daily production quantity $x$ does the profit $L(x)$ reach its maximum value, and what is this maximum value?  
(2) At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?  
(3) Due to an economic crisis, the factory had to lay off workers, causing the maximum daily production capacity to drop to 160 units. At what daily production quantity $x$ does the average profit $P(x)$ reach its maximum value, and what is this maximum value?

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