
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
With the gradual implementation of the "double reduction" policy, the number of registrations for the visually impaired sports club at a certain school has surged. The club has decided to purchase a certain number of basketballs and soccer balls from a sports equipment store for daily training. It is known that the price of each basketball is $30$ yuan higher than the price of each soccer ball. The number of soccer balls that can be purchased with $1500$ yuan is the same as the number of basketballs that can be purchased with $2400$ yuan.  
$(1)$ Find the unit price of basketballs and soccer balls.  
$(2)$ Based on the student registration situation, the club needs to purchase a total of $100$ basketballs and soccer balls at once, with the condition that the number of soccer balls purchased does not exceed $\frac{1}{3}$ of the number of basketballs purchased. During the actual purchase, the price of each basketball is reduced by $m$ yuan ($25\leqslant m\leqslant 35$). Find the minimum value of the actual purchase cost. (The result can be expressed as an algebraic expression involving $m$).

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