
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 certain company's annual fixed cost for producing a certain electronic instrument is $20$ million yuan. When the annual output is $x$ thousand units, an additional cost $C\left(x\right)$ (in million yuan) is required. $C(x)=\left\{\begin{array}{c}\frac{1}{2}{x}^{2}+10x+1100,0＜x＜100,\\ 120x+\frac{4500}{x-90}-5400,x≥100\end{array}\right.$ The selling price of each product is $100$ million yuan. For simplicity, we assume that all products produced by the company can be sold.<br/>$(1)$ Write the annual profit $L$ (in million yuan) as a function of the annual output $x$ (in thousand units);<br/>$(2)$ For what value of the annual output in thousand units does the company achieve maximum profit? What is the maximum profit?

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