
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 the mascot of the Qatar World Cup. Through market research, it was found that the monthly sales quantity $y$ (in units) of this product is a linear function of the selling price $x$ (in dollars per unit). The corresponding values of the selling price $x$ and the monthly sales quantity $y$ are shown in the table below:

| Selling Price $x$ (dollars per unit) | $30$ | $45$ |
|--------------------------------------|------|------|
| Monthly Sales Quantity $y$ (units)   | $300$| $270$|

$(1)$
1. Find the expression of $y$ in terms of $x.
2. The cost price of this product is $30$ dollars. For what selling price $x$ will the monthly sales profit $w$ (in dollars) be maximized? Calculate the maximum profit. [$Note$: Monthly sales profit $=$ Monthly sales quantity $\times$ (Selling price $-$ Cost price)]

$(2)$
For the selling price at which the profit is not less than $10000$, what is the minimum selling price needed?

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