
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
In the activity of building a beautiful countryside, a village committee is planning to plant cypress and fir trees on both sides of the rural roads. According to a market survey, it is found that buying $2$ cypress trees and $3$ fir trees costs a total of $440$ yuan, and buying $3$ cypress trees and $1$ fir tree costs a total of $380$ yuan. <br/>$(1)$ Find the unit price of cypress and fir trees; <br/>$(2)$ If a total of $150$ trees (both types of trees must be purchased) are to be purchased to beautify the rural roads this time, and the number of cypress trees is not less than three times the number of fir trees, let the number of cypress trees purchased be $x$ in this activity, and the cost of purchasing trees this time be $w$ yuan. <br/>① Find the functional expression between $w$ and $x$, and write down the range of values for $x$? <br/>② To minimize the cost of purchasing trees this time, how many trees of each type should be purchased? What is the minimum cost in yuan?

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