
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
To revitalize the nation and rejuvenate the countryside, cooperatives play a crucial role in the revitalization of rural industries. Farmer professional cooperatives have become a new type of agricultural entity and a key force in modern agricultural development, making significant contributions to the implementation of the rural revitalization strategy. A farmer professional cooperative is engaged in processing a certain brand of clothing. It is known that a fixed cost of $30$ thousand yuan is required annually for processing this brand of clothing. For processing each $x$ thousand pieces of this brand of clothing, an additional cost of $f(x)$ thousand yuan is needed, where $f(x)=\left\{\begin{array}{l}\frac{1}{2}{x}^{2}+2x,0＜x≤10,\\ 14x+\frac{450}{x}-115,10＜x≤50.\end{array}\right.$ According to market conditions, the cooperative earns $12$ yuan as processing fee for each piece of clothing processed.<br/>$(1)$ Find the analytical expression of the annual profit $y$ (in thousand yuan) of the cooperative for processing this brand of clothing as a function of the annual processing quantity $x$ (in thousand pieces).<br/>$(2)$ For how many thousand pieces of clothing processed annually will the cooperative earn the maximum profit for processing this brand of clothing? Also, determine the maximum profit earned annually.

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