
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 2022, a certain company charges $25$ yuan per ton for kitchen waste disposal and $16$ yuan per ton for construction waste disposal. The company paid a total of $5200$ yuan for the disposal of kitchen and construction waste. Starting from January 2023, the charging standards are adjusted to $100$ yuan per ton for kitchen waste disposal and $30$ yuan per ton for construction waste disposal. If the quantities of these two types of waste handled by the company in 2023 remain unchanged compared to 2022, the company will need to pay an additional $8800 for waste disposal.  
$(1)$ How many tons of kitchen waste and construction waste did the company handle in 2022?  
$(2)$ The company plans to reduce the total amount of the two types of waste disposal to 240 tons in 2023, with the amount of construction waste disposal not exceeding three times the amount of kitchen waste disposal. How much will the company need to pay for the disposal of these two types of waste in 2023 at least?

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