
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
After long-term development, China has successfully embarked on a poverty alleviation path with Chinese characteristics. A rural area is dedicated to building a "special ecological fruit base" according to local conditions. Through research, it is found that the yield $L$ (in kilograms) of a rare fruit tree is related to the amount of fertilizer $x$ (in kilograms) by the function: $L(x)=\left\{\begin{array}{l}5({{x^2}+6}),0≤x≤2\\ \frac{{75x}}{{1+x}},2＜x≤5\end{array}\right.$. The fertilizer cost for each fruit tree is $20x$ yuan, and other costs (such as cultivation management, labor for fertilization, etc.) are $25x$ yuan. It is known that the market price of this fruit is approximately $15$ yuan per kilogram, and the demand exceeds supply. Let the profit of each fruit tree be denoted as $f(x)$ (in yuan).  
$(1)$ Find the functional relationship of $f(x)$;  
$(2)$ For how many kilograms of fertilizer per tree, the profit of the fruit tree is maximized? What is the maximum profit?

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