
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
An enterprise is preparing to invest in two productive projects, $A$ and $B$. According to the analysis of factors such as production costs and sales, it is known that 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 $y_{A}=\frac{2}{5}x$, 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 $y_{B}=-\frac{1}{5}x^{2}+2x$.

$(1)$ If $10$ thousand yuan is invested in project $A$, how much profit will be obtained after one year?

$(2)$ If the same amount of funds $m$ (where $m>0$) thousand yuan is invested in projects $A$ and $B, and the profits obtained by both are equal after one year, what is the value of $m$?

$(3)$ In 2023, China implements an income tax preferential policy for small and micro enterprises. The enterprise will invest a total of $32$ thousand yuan in tax relief and other surplus funds into projects $A$ and $B. When investing in projects $A$ and $B, how much profit will be maximized one year later if the funds are allocated optimally? What is the maximum value in thousand yuan?

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