
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
To solidly promote the work of "Five Education" and strengthen labor education, a school spent $2000$ yuan to purchase a batch of type $A$ labor tools. After some time, they needed to buy a second batch of type $A$ labor tools, with the price of each tool increasing by $5$ yuan. The school spent $2200$ yuan to purchase the same quantity of type $A$ labor tools as the first batch.  
$(1)$ What is the price per tool of the first batch of type $A$ labor tools purchased by the school?  
$(2)$ If the school needs to purchase a total of $50$ tools in the third batch, with the price per tool of type $A$ labor tools the same as the second batch, and each type $B$ labor tool costing $40$ yuan, planning to spend no more than $2500$ yuan on both type $A$ and type $B$ labor tools, how many type $A$ labor tools can be purchased at most?

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