
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
As a key force in implementing the national strategy of "rural revitalization," the supply and marketing cooperative can help farmers distribute and coordinate agricultural products, promote the quick construction of a unified national large market, and bring real benefits to the people. A certain supply and marketing cooperative guides farmers in producing and selling local specialties. They conducted a series of analyses on the production, market demand, cost, and selling price of this specialty product. They found that the production of this specialty product $y_{production}$ (unit: tons) is a linear function of the selling price $x$ (unit: yuan per kilogram), that is, $y_{production}=200x-100$. The market demand $y_{demand}$ (unit: tons) for this product is a quadratic function of the selling price $x$ (unit: yuan per kilogram), with some corresponding values as shown in the table below:

| Selling Price $x$ (yuan per kilogram) | ... | $2$ | $3$ | $4$ | $5$ | ... |
|---------------------------------------|-----|-----|-----|-----|-----|-----|
| Demand $y_{demand}$ (tons)            | ... | $1020$ | $1020$ | $980$ | $900$ | ... |

It was also found that the selling price $x$ (unit: yuan per kilogram) and cost $z$ (unit: yuan per kilogram) of this specialty product vary with time $t$ (months). Their functional expressions are $x=t+1$ and $z=\frac{1}{8}t^{2}+\frac{3}{2}$.

$(1)$ Write down the analytical expression of the market demand $y_{demand}$ as a function of the selling price $x$ (no need to specify the range of the independent variable).

$(2)$ In which month does the profit per kilogram of selling this specialty product reach its maximum? What is the maximum value?

$(3)$ The supply and marketing cooperative plays a functional role in avoiding waste and guiding farmers in production. If the production quantity of this specialty product is exactly equal to the market demand, calculate the total profit obtained from selling all the specialty products at that time.

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