
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 plans to install warm reminder signs for garbage classification and garbage bins. If the school purchases $2$ warm reminder signs and $3$ garbage bins for a total of $550$ yuan, and the price of a garbage bin is $3$ times the price of a warm reminder sign.<br/>$(1)$ Find the price of a warm reminder sign and a garbage bin, respectively.<br/>$(2)$ The school needs to place at least $48$ garbage bins. If the school purchases a total of $100$ warm reminder signs and garbage bins, and the cost does not exceed $10000$ yuan, please list all the purchasing plans, and indicate which plan requires the least amount of money, and how much is the minimum amount?

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