
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 demand of citizens for fruits, a fruit shop purchased apples and pears at prices of $15$ yuan per kilogram and $10$ yuan per kilogram respectively, totaling 200 kilograms. The apples were sold at a profit of $40\%$ per kilogram, and the selling price of pears per kilogram was $\frac{2}{3}$ of the selling price of apples per kilogram. After a period of time, both types of fruits were sold out, and it was found that the total profit from selling these two types of fruits was 1020 yuan.  
$(1)$ How many kilograms of apples and pears did the fruit shop purchase this time?  
$(2)$ Because there was still demand for these two types of fruits, the fruit shop purchased some apples and pears at the same prices as before. The quantity of apples purchased this time was 10 kilograms less than the previous purchase, while the quantity of pears purchased remained the same. Due to market reasons, the selling price per kilogram of apples was reduced by $\frac{3}{5}a\%$, and the selling price per kilogram of pears was increased by $\frac{2}{5}a\%$. If the total profit from selling these apples and pears must not be less than 771 yuan, find the maximum value of $a$.

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