
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 innovative technology company, in response to the call of the municipal government, decided to research and develop a new type of industrial robot. After market research, it was found that the annual fixed cost for producing robots is $1$ million yuan. For each production of $x$ robots, an additional variable cost of $C\left(x\right)$ million yuan is incurred. When the annual output is less than $80$ units, $C(x)=\frac{1}{{30}}{x^2}+2x$ (million yuan); when the annual output is not less than $80$ units, $C(x)=\frac{{103}}{{17}}x+\frac{{425}}{x}-135$ (million yuan). The selling price of each industrial robot is $6$ million yuan. Through market analysis, it is determined that all produced robots can be sold in the same year.<br/>$(1)$ Write the analytical expression of the function of annual profit $L\left(x\right)$ (million yuan) with respect to the annual output $x$ (units). (Note: Annual profit $=$ Annual sales revenue $-$ Fixed cost $-$ Variable cost)<br/>$(2)$ For what annual output is the profit maximized in the production of industrial robots? What is the maximum profit?

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