
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
In recent years, the food delivery industry in China has been developing rapidly, and delivery drivers shuttling through the streets of cities have become a beautiful scenery. A certain food delivery driver travels to and from $4$ different food delivery stores (numbered $1, 2, 3, 4$) every day. The rule is: he first picks up an order from store $1$, called the first pick-up, and then he goes to any of the other $3$ stores for the second pick-up, and so on. Assuming that starting from the second pick-up, he always goes to one of the other $3$ stores that he did not pick up from last time. Let event $A_{k}=\{$the $k$-th pick-up is exactly from store $1\}$, $P(A_{k})$ is the probability of event $A_{k}$ occurring. Obviously, $P(A_{1})=1$, $P(A_{2})=0$. Then $P(A_{3})=$______, $P(A_{10})=$______ (round the second answer to $0.01$).

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