
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 mobile phone in the year 2024 in order to further increase its market competitiveness. Through market research, it is found that the annual research and development cost required to produce this product is 2.5 million yuan. For each production of $x$ (thousand units) of the phone, an additional cost of $R(x)$ million yuan is required, where $R(x)=\left\{{\begin{array}{l}{10{x^2}+100x+800,0<x<50,}\\{504x+\frac{{10000}}{{x-2}}-6450,x\geq 50,}\end{array}}\right.$ 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/>$(Ⅰ)$ Find the function relationship of the profit $y$ of this phone in 2024 with the annual output $x$;<br/>$(Ⅱ)$ For what annual output $x$ does the company obtain the maximum profit in that year? What is the maximum profit?

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