
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 order to inherit the spirit of Lei Feng, a certain middle school launched a "donate with love" fundraising activity to all teachers and students in the school, preparing to donate basketballs and soccer balls to schools in the western mountainous areas. It is known that the unit price of a basketball is $100$ yuan each, and the unit price of a soccer ball is $80$ yuan each.  
$(1)$ The original plan was to raise $5600$ yuan, all of which would be used to purchase basketballs and soccer balls. If exactly $60$ items could be purchased, how many basketballs and soccer balls should be bought respectively?  
$(2)$ In the donation activity, due to the high enthusiasm of the teachers and students, a total of $6890$ yuan was actually received. If a total of $80$ basketballs and soccer balls were purchased, and the expenditure did not exceed $6890$ yuan, how many basketballs could be bought at most?

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