
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 school has two restaurants, $A$ and $B$. It is known that a student, let's call him student $A$, will choose one of these two restaurants for lunch every day. If student $A$ chooses a certain restaurant on a given day, there is a $60\%$ chance that he will switch to the other restaurant for the next day. If on the first day student $A$ chooses restaurant $A$, then the probability of choosing restaurant $A$ on the $n$-th day is ______.

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