
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
This semester, the school cafeteria conducted a survey on the staple food purchasing situation of teachers and students (only rice and noodles were served as staples, with the number of diners remaining stable). The survey found that of those who bought rice, 20% would switch to noodles the next time, and of those who bought noodles, 30% would change to rice the next time. Let $a_n$ and $b_n$ represent the proportions of people who buy rice and noodles, respectively, on the nth purchase. Assume the proportions were exactly equal the first time, i.e., $a_{1}=b_{1}= \frac{1}{2}$.
(1) Find the value of $a_n+b_n$.
(2) Write the recursion formula for the sequence $\{a_n\}$.
(3) Find the general formula of the sequences $\{a_n\}$ and $\{b_n\}$ and state how many servings of rice and noodles the cafeteria should prepare, approximately, to satisfy all the teachers and students (assuming there are 2000 diners), as time goes by.

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