
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
Hameijia School from another school is planning to purchase two brands of soccer balls, brand $A$ and brand $B$, to welcome the "2023 New Year's Day Snow Football Match". It is known that the unit price of brand $A$ soccer balls is $10$ yuan more than brand $B$ soccer balls. If $20$ brand $A$ soccer balls and $15$ brand $B$ soccer balls cost $3350$ yuan, find:<br/>
$(1)$ The price of each brand $A$ soccer ball and each brand $B$ soccer ball;<br/>
$(2)$ Hameijia School from another school decides to purchase a total of $50$ brand $A$ and brand $B$ soccer balls, with a total cost not exceeding $4650$ yuan. How many brand $A$ soccer balls can they purchase at most?

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