
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 jar A, there are 5 red balls, 2 white balls, and 3 black balls. In jar B, there are 4 red balls, 3 white balls, and 3 black balls. First, randomly take a ball from jar A and put it into jar B. Let $A_1$, $A_2$, and $A_3$ represent the events that the ball taken out from jar A is a red ball, a white ball, and a black ball, respectively. Then, randomly take a ball from jar B, and let event B represent the event that the ball taken out from jar B is a red ball. Among the following conclusions, the correct ones are __________ (write down all the correct conclusion numbers).
① $P(B)=$； ② $P(B|A_1)=$； ③ Event B and event $A_1$ are independent；④ $A_1$, $A_2$, $A_3$ are mutually exclusive events；  
⑤ The value of $P(B)$ cannot be determined because it depends on which one of $A_1$, $A_2$, $A_3$ occurs.

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