
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 implement the environmental concept of "garbage classification," a certain school purchased green and gray garbage bins at the same time. If they buy $2$ green garbage bins and $1$ gray garbage bin, it will cost a total of $280$ yuan; if they buy $3$ green garbage bins and $2$ gray garbage bins, it will cost a total of $460$ yuan.  
$(1)$ Find the purchase price of each green and gray garbage bin.  
$(2)$ To establish a model school for garbage classification, the school plans to use no more than $9000$ yuan to purchase a total of $100$ bins of two colors, with the number of green bins not less than $80\%$ of the number of gray bins. How many purchasing plans are there?  
$(3)$ For each green and gray garbage bin purchased, the government will subsidize $m$ yuan and $n$ yuan, respectively. If the costs of all purchasing plans in $(2)$ are the same, find the relationship between $m$ and $n$.

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