
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 $100$ dollars per item, and it is stipulated that the selling price must not exceed $1.4$ times the cost price. A market survey revealed 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) | $130$ | $140$ |
|--------------------------------------|-------|-------|
| Sales Quantity $y$ (items per day)   | $140$ | $120$ |

$(1)$ Write down the function relationship of $y$ with respect to $x$ directly.  
$(2)$ Let the profit obtained by the store from selling this product daily be $W$ (in dollars). Find the function relationship between $W$ and $x$, and determine at what selling price the store will maximize the profit from selling this product daily.

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