
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 plans to use new technology to produce a new mobile phone in the year $2023$ in order to further increase its market competitiveness. Through market research, it is found that the production of this product requires a research and development cost of $2.5$ million yuan for the whole year. For each production of $x$ (thousand units) of the phone, an additional cost of $R(x)$ million yuan is required, where $R(x)=\left\{\begin{array}{c}10{x}^{2}+100x+800,0<x<50\\ 504x+\frac{10000}{x-2}-6450,x\geq50\end{array}\right.$. It is known that the selling price of each phone is $5000$ yuan, and all the phones produced are sold out in the same year.<br/>$(1)$ Find the function relationship $y$ of the profit of this phone in $2023$ with respect to the annual output $x$;<br/>$(2)$ For what annual output $x$ does the company obtain the maximum profit? What is the maximum profit?

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