
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
Person A and person B process the same type of component. The quantity processed by person A each day is $1.5$ times the quantity processed by person B each day. Both of them process $600$ of these components each. Person A takes $5$ days less than person B to complete the task.<br/>$(1)$ Find out how many of these components person A and person B process each day respectively.<br/>$(2)$ It is known that the daily processing fees for person A and person B for these components are $150$ yuan and $120$ yuan respectively. There is a task to process $3000$ of these components. After processing alone for a period of time, person A has other arrangements, and the remaining task is completed by person B alone. If the total processing fee does not exceed $7800$ yuan, then how many days did person A work at least?

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