
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
With the arrival of summer, a bedding store is preparing to introduce two different models of bamboo mats, $A$ and $B$. When purchasing 10 pieces of model $A$ and 15 pieces of model $B$, the cost is 3600 yuan. When purchasing 25 pieces of model $A$ and 30 pieces of model $B$, the cost is 8100 yuan.  
$(1)$ Find the purchase price per piece of model $A$ and model $B$ bamboo mats.  
$(2)$ It is known that the selling price of model $A$ bamboo mat is 260 yuan per piece, and the selling price of model $B$ bamboo mat is 180 yuan per piece. Now, preparing to purchase a total of 60 pieces of model $A$ and model $B$ bamboo mats, if the total profit after selling all of them is not less than 4400 yuan, find the minimum number of model $A$ bamboo mats that need to be purchased.

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