
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
Qinzhou Huangxiaomi, originally named "Cao Gu" or "Pashan Cao," was bestowed the name "Qinzhou Huang" by the Qing Kangxi Emperor. It has long been famous for its royal tribute rice, representing Shanxi millet and enjoying the titles of "King of Rice in the World" and "National Rice." Qinzhou Huangxiaomi has a waxy yellow color, crystal clear, round grains, and a pearl-like shape. A folk saying goes, "Golden pearls do not exchange for Qinzhou Huang." After research, it was found that the yield per mu of Qinzhou Huangxiaomi $T$ (unit: kg) and the amount of fertilizer applied $x$ (unit: kg) satisfy the function relationship: $T=\left\{\begin{array}{l}\frac{{x}^{2}}{20}+50,0＜x≤25,\\ \frac{91x}{3+x},25＜x≤50,\end{array}\right.$ and the cost of fertilizer is $5$ yuan per kg, and the labor cost for fertilization is $1$ yuan per kg. It is known that the market price of Qinzhou Huangxiaomi is $30$ yuan per kg, and the demand exceeds supply. Let the profit of one mu of Qinzhou Huangxiaomi be $f\left(x\right)$ (unit: yuan).<br/>$(1)$ Find the analytical expression of $f\left(x\right)$;<br/>$(2)$ This year, Farmer Bo planted a total of $5$ mu of Qinzhou Huang rice. When the amount of fertilizer applied per mu of land is how many kg (rounded to $1$ kg), will Farmer Bo's profit be maximized? What is the maximum profit? (rounded to $1$)<br/>Reference data: $\sqrt{6×8190}≈222,\sqrt{1365}≈37$

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