
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 middle school is preparing to purchase two types of experimental equipment, $A$ and $B$. Below are two sales records presented by the salesperson (the unit price of these two types of experimental equipment remains the same for each sale), as shown in the table:

|         | $A (\text{pieces})$ | $B (\text{pieces})$ | Amount (yuan) |
|---------|---------------------|---------------------|---------------|
| First   | $20$                | $10$                | $1100$        |
| Second  | $25$                | $20$                | $1750$        |

$(1)$ Find the unit prices of type $A$ and type $B$ experimental equipment in yuan.  
$(2)$ If a total of $50$ pieces of these two types of experimental equipment are purchased, with the quantity of type $A$ equipment (in pieces) not exceeding twice the quantity of type $B$ equipment (in pieces), and the total cost does not exceed $2000$ yuan, how many purchasing plans are there?

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