
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 is actively preparing for the "Sunshine Sports" activity and has decided to purchase a batch of basketballs and soccer balls totaling $30$ items. At a sports equipment store, each basketball costs $80$ yuan, and each soccer ball costs $60$ yuan. During the purchase period at the school, there is a promotion for soccer balls at 20% off. Let $m\left(0 \lt m \lt 30\right)$ be the number of basketballs the school wants to purchase, and let $w$ be the total cost of purchasing basketballs and soccer balls.<br/>$(1)$ The analytical expression of the function between $w$ and $m$ is ______;<br/>$(2)$ If the school requires the number of basketballs purchased to be at least twice the number of soccer balls, then the school will spend the least amount when purchasing ______ basketballs, and the minimum value of $w$ is ______ yuan.

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