
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
A company is developing a new product using two types of raw materials, $A$ and $B$, with $360$ kilograms and $290$ kilograms of each, respectively. They are producing two new products, labeled as type A and type B, with a total of $50$ units. The table below shows the amount of raw materials required for each unit of the new products:

| Raw Material Content | Product A ($kg$) | Product B ($kg$) |
|----------------------|------------------|------------------|
| A                    | $9$              | $3$              |
| B                    | $4$              | $10$             |

$(1)$ If $x$ units of type A products are produced, write down the system of inequalities based on the given conditions to determine the range of values for $x$. <br/> $(2)$ If the cost of producing one unit of type A product is $70$ yuan and one unit of type B product is $90$ yuan, and the total cost of producing both products is $y$ yuan, write down the function relationship between the total cost $y$ (in yuan) and the number of units of type A products $x$. When a certain number of units of type A and type B products are produced, what is the minimum total cost of production?

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