
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 store sells a product with a cost price of $50$ dollars per item. Through market research, it was found that the daily sales quantity $y$ (in items) of the product is a linear function of the selling price $x$ (in dollars per item). The corresponding values of selling price and sales quantity are shown in the table below:

| Selling Price $x$ (dollars per item) | $55$ | $65$ |
|--------------------------------------|------|------|
| Sales Quantity $y$ (items per day)   | $90$ | $70$ |

$(1)$ If the profit from sales on a certain day is $800$ dollars, what is the selling price per item on that day?

$(2)$ Let $W$ be the profit obtained by the store from selling the product each day. Find the functional relationship between $W$ and $x$, and determine at what selling price the store maximizes its daily profit from selling this product.

$(3)$ Due to certain reasons, the cost price of the product increased by $a$ dollars per item $(a > 0)$. In the future sales, the daily sales quantity and selling price of the product still satisfy the original functional relationship. It is specified that the selling price should not be lower than the cost price and should not exceed $70$ dollars per item. If the maximum profit that can be obtained each day in the future is $960$ dollars, find the value of $a$.

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