
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 supermarket sells two models of appliances, model A and model B, with purchase prices of $180$ yuan per unit and $160$ yuan per unit, respectively. The table below shows the sales situation in the past two weeks (purchase price and selling price remain unchanged, profit $=$ selling price $-$ purchase price):

| Sales Period | Quantity Sold (Model A) | Quantity Sold (Model B) | Sales Revenue |
|--------------|-------------------------|-------------------------|---------------|
| Week 1       | 3                       | 2                       | $1120         |
| Week 2       | 4                       | 3                       | $1560         |

$(1)$ Find the selling prices of models A and B.  
$(2)$ If the supermarket is prepared to purchase a total of 35 units of these two models of appliances with no more than $6000$ yuan, how many units of model A can be purchased at most?  
$(3)$ Under the condition in $(2)$, can the supermarket achieve a profit goal exceeding $1750$ yuan after selling these 35 units of appliances? If yes, provide the corresponding purchase plan and indicate which plan yields the maximum profit. If not, provide the reason.

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