
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
A certain factory produced $1250$ liters of a new disinfectant in December of last year, with a pass rate of $90\%$. Starting from January of this year, the factory will officially produce this disinfectant for the next two years. In January of this year, the production volume and pass rate will be the same as in December of last year. From then on, the production volume will increase by $5\%$ each month based on the previous month, and the pass rate will increase by $0.4\%$ compared to the previous month.<br/>$(1)$ Calculate the total production volume of this disinfectant from January to December of this year; (rounded to $1$ liter)<br/>$(2)$ From which month onwards can the amount of non-conforming disinfectant produced per month be kept within $100$ liters?

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