
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
To meet the shopping needs of customers during the "May Day" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is $6 yuan per kilogram.  
$(1)$ Find out how many kilograms of cherries and cantaloupes the fruit store purchased.  
$(2)$ After selling the first batch of purchased fruits, the fruit supermarket decided to reward customers by launching a promotional activity. They purchased a total of $300$ kilograms of cherries and cantaloupes for no more than $5280$ yuan. Among them, $a$ kilograms of cherries and $2a$ kilograms of cantaloupes were sold at the purchase price, while the remaining cherries were sold at $55$ yuan per kilogram and cantaloupes were sold at $10$ yuan per kilogram. If the supermarket owner plans to make a profit of at least $2120$ yuan in this transaction, find the maximum value of the positive integer $a$.

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