
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 group has discovered that the yield of a nectarine tree, denoted as $w$ (in units of hundred kilograms), is related to the fertilizer cost, denoted as $x$ (in units of hundred yuan), as follows:

$$w=4-\frac{3}{x+1}$$

The fertilizer cost does not exceed 5 hundred yuan. Additionally, other costs (such as labor costs for fertilization) amount to $2x$ hundred yuan. The market price of this variety of nectarine is 16 yuan per kilogram (or 16 hundred yuan per hundred kilograms), and the market demand is always greater than the supply. Denote the profit of this nectarine tree as $L(x)$ (in units of hundred yuan).

(1) Find the profit function $L(x)$ and write out its domain;

(2) When the fertilizer cost is optimal, what is the maximum profit?

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