
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
In recent years, in the context of the Internet of Things, industrial "digital twin" technology has become a research hotspot. It uses digital technology to map the information of objects, systems, and processes in real time, completes virtual simulation processes, and significantly reduces the heavy asset and high cost burden faced in the research of technological innovation and decision optimization in the industrial field. A certain company is planning to invest in two productive projects, $A$ and $B$, using "digital twin" technology. An analysis based on factors such as production costs and sales reveals the following: The function expression of the profit $y_{A}$ (in ten thousand yuan) one year after investing in project $A$ with an input of funds $x$ (in ten thousand yuan) is $y_{A}=2x$, and the function expression of the profit $y_{B}$ (in ten thousand yuan) one year after investing in project $B$ with an input of funds $x$ (in ten thousand yuan) is $y_{B}=-x^{2}+10x$.  
$(1)$ If $10$ thousand yuan is invested in project $A$, what is the profit one year later?  
$(2)$ If the same amount of funds $m$ (where $m>0$) is invested in projects $A$ and $B$, and the profits obtained from both projects are equal one year later, what is the value of $m$?  
$(3)$ In 2023, China implements a preferential income tax policy for small and micro enterprises. The company will receive a total of $32$ thousand yuan in tax relief and other surplus funds, all of which will be invested in projects $A$ and $B$. When investing in projects $A$ and $B$ with how many thousand yuan each, the sum of the profits obtained one year later is maximized? What is the maximum value in thousand yuan?

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