
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
The development of new energy vehicles worldwide is advancing rapidly. Electric vehicles are mainly divided into three categories: pure electric vehicles, hybrid electric vehicles, and fuel cell electric vehicles. These three types of electric vehicles are currently at different stages of development and each has its own development strategy. China's electric vehicle revolution has long been underway, replacing gasoline and diesel vehicles with new energy vehicles. China is vigorously implementing a plan that will reshape the global automotive industry. In the year 2022, a certain company plans to introduce new energy vehicle production equipment. Through market analysis, it is determined that a fixed cost of 20 million yuan needs to be invested throughout the year. For each production of x (hundreds of vehicles), an additional cost C(x) (in million yuan) needs to be invested, and C(x) is defined as follows: $C(x)=\begin{cases}10x^2+100x, & 0<x<40,\\501x+\frac{10000}{x}-4500, & x\geq40\end{cases}$. It is known that the selling price of each vehicle is 50,000 yuan. According to market research, all vehicles produced within the year can be sold. 

$(1)$ Find the function relationship of the profit $L(x)$ in the year 2022 with respect to the annual output $x$ (in hundreds of vehicles).

$(2)$ For the year 2022, at what output level in hundreds of vehicles will the company achieve the maximum profit? Calculate the maximum profit.

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