
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 takeout industry in China has developed rapidly, and takeout delivery drivers shuttle through the streets and alleys of the city, becoming a beautiful scenery. They pick up orders from takeout restaurants based on the information provided by the takeout platforms. A certain takeout delivery driver travels between $10$ takeout restaurants (numbered $1, 2, \cdots, 10$) every day. It is agreed that he first picks up an order from restaurant $1$, called the first pick-up, and then, he goes to any of the other $9$ restaurants for the second pick-up, and so on. Assuming that starting from the second pick-up, he always picks up orders from one of the other $9$ restaurants not visited in the previous pick-up, let $A_{k}=\{$the $k$-th pick-up is exactly from restaurant $1\}$, $P(A_{k})$ 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}
user