
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 research team found that the yield of a peach tree, $\omega$ (in kilograms), and the cost of fertilizer, $x$ (in hundred yuan), satisfy the following relationship: $\omega = 4 - \frac{3}{x+1}$, and the cost of fertilizer does not exceed 500 yuan. In addition, other costs of 2$x$ (such as labor costs, etc.) in hundred yuan are also required. Given that the market price of these peaches is 16 yuan/kg (i.e., 1600 yuan per 100 kg), and the market demand always exceeds supply. Let the profit obtained from this peach tree be $L(x)$ (in hundred yuan).
(1) Find the expression for the profit function $L(x)$, and state its domain.
(2) At what fertilizer cost is the profit from this peach tree maximized? What is the maximum profit?

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