
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]


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A certain factory plans to choose one of two products, $A$ and $B$, for production and sales, with a daily production and sales quantity of $x$ units. It is known that the cost price of product $A$ is $m$ yuan per unit ($m$ is a constant, and $4 \leq m \leq 6$), the selling price is 8 yuan per unit, with a maximum daily production and sales quantity of 500 units, and a daily patent fee of 30 yuan. The cost price of product $B$ is 12 yuan per unit, the selling price is 20 yuan per unit, with a maximum daily production and sales quantity of 300 units, and a daily patent fee of $y$ yuan, where $y$ (in yuan) and the daily production and sales quantity $x$ (in units) satisfy the relationship $y = 80 + 0.01x^{2}$.

$(1)$ If the daily profits from the production and sales of products $A$ and $B$ are $w_{1}$ yuan and $w_{2}$ yuan, respectively, write down the functional relationships of $w_{1}$, $w_{2}$ with $x$, and specify the range of values for $x$.

$(2)$ Calculate the maximum daily profits from the production and sales of products $A$ and $B$. (Express the maximum daily profit of product $A$ using an algebraic expression involving $m$).

$(3)$ To achieve the maximum daily profit, which product should the factory choose for production and sales? Provide a justification.

【Profit = (Selling Price - Cost) × Quantity Sold - Patent Fee】

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