
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
Our city is striving to become a national sanitary city. A certain street is actively responding and has decided to install warm reminder signs for garbage classification and garbage bins in all residential areas on the street. If buying $4$ garbage bins costs $350$ yuan more than buying $5$ warm reminder signs, and the price of a garbage bin is three times the price of a warm reminder sign.<br/>$(1)$ Find the price of a garbage bin and a warm reminder sign, respectively.<br/>$(2)$ If the street needs to purchase a total of $3000$ warm reminder signs and garbage bins.<br/>① Find the functional relationship between the cost $w$ (in yuan) of purchasing warm reminder signs and garbage bins and the number of warm reminder signs $m$;<br/>② If the street plans to spend no more than 350,000 yuan on purchasing warm reminder signs and garbage bins, and the number of garbage bins is at least $1.5$ times the number of warm reminder signs, then: When the street purchases how many warm reminder signs, the cost is the most economical? What is the most economical cost?

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