
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 certain enterprise in our city arranges 65 workers to produce two types of products, A and B. Each worker can produce 2 units of product A or 1 unit of product B per day. Through testing, it is found that the profit for each unit of product A is $15, and for each unit of product B is $120. However, in actual production, producing product B requires additional expenses. After calculation, it is found that for each unit of product B produced, the daily average production cost increases by $2 per unit (profit decreases). Let $x$ represent the number of workers assigned to produce product B each day.

$(1)$ Find the functional relationship between the profit $y_{1}$ (in dollars) from producing product A and the profit $y_{2}$ (in dollars) from producing product B each day, with respect to $x$.

$(2)$ If the profit from producing product A is $1250 more than the profit from producing product B each day, find the value of $x$.

$(3)$ Let $W$ represent the total profit from producing both products A and B. Find the maximum value of $W$ and the corresponding value of $x$.

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