
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
The school cafeteria serves 1000 students daily, offering two options, A and B, every Monday. Of those who choose option A on a given Monday, 20% opt for option B the following Monday. Conversely, 30% of those who choose option B switch to option A the following Monday. Let A$_{
m n}$ and B$_{
m n}$ represent the number of students choosing options A and B, respectively, on the nth Monday. Calculate $$\overset{lim}{n\rightarrow \infty }A_{n}$$.

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