
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 $LED$ lamps have the functions of energy saving and environmental protection, and have a long service life. According to market research, it is known that the annual fixed cost required to produce a certain type of $LED$ lamp is $40,000$ yuan for every $x$ thousand units of the product produced. An additional variable cost of $W(x)$ yuan is required to produce $x$ thousand units of the product. When the annual output is less than $60,000$ units, $W(x)=\frac{1}{2}{x^2}+x$. When the annual output is not less than $60,000$ units, $W(x)=7x+\frac{{100}}{x}-39$. The selling price of each product is $6$ yuan. Assume that the annual sales volume of the product is equal to the annual output.<br/>$(Ⅰ)$ Write the analytical expression of the function of annual profit $L(x)$ in terms of the annual output $x$ (in thousand units). (Note: Annual profit $=$ Annual sales revenue $-$ Fixed cost $-$ Variable cost)<br/>$(Ⅱ)$ For how many thousand units of annual output is the annual profit maximized? What is the maximum annual profit?

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