
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
During the Spring Festival, a fruit shop purchased 100 kilograms of peaches and 50 kilograms of apples. The cost price of apples is 1.2 times the cost price of peaches. Peaches are sold at a price of $16 per kilogram, and apples are sold at a price of $20 per kilogram. Both types of fruits were sold out on that day, and the fruit shop made a profit of $1800.<br/>$(1)$ Find the cost price of peaches per kilogram.<br/>$(2)$ After the first batch of peaches was sold out, the fruit shop purchased another 300 kilograms of peaches at the same cost price. Seeing that the first batch of fruits sold well, the shop increased the price of peaches to $17 per kilogram on the first day and sold 8a kilograms. Due to the difficulty in preserving peaches, the shop reduced the price by $0.1a per kilogram from the original price of $16 per kilogram on the second day. By the time the shop closed in the evening, there were still 20 kilograms left unsold. The owner distributed the remaining peaches to the staff, and the profit from this batch of peaches was $2980. Find the value of $a$.

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