
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 school is preparing to purchase sports teaching equipment $A$ and $B$. The table below shows the price information for these two types of equipment:

| $A$ (units) | $B$ (units) | Total Cost |
|-------------|-------------|------------|
| 3           | 1           | 500 yuan   |
| 1           | 2           | 250 yuan   |

$(1)$ (Solving applied problems using a system of two linear equations) Find the selling prices of each unit of equipment $A$ and equipment $B$.  
$(2)$ If the school is prepared to spend no more than 2700 yuan to purchase a total of 25 units of these two types of equipment, how many units of equipment $A$ can be purchased at most?  
$(3)$ Under the condition in $(2)$, purchasing a total of 25 units of these two types of equipment with at least 12 units of equipment $A$, what are the possible purchasing plans, and what is the minimum cost?

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