
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
Bag A contains 5 red balls, 2 white balls, and 3 black balls. Bag B contains 4 red balls, 3 white balls, and 3 black balls. A ball is randomly drawn from Bag A and placed into Bag B. Let $A_1$, $A_2$, and $A_3$ denote the events that the ball drawn from Bag A is red, white, and black, respectively. Then, a ball is randomly drawn from Bag B; let B represent the event that the ball drawn from Bag B is red. Which of the following conclusions are correct: ①$P(B) = \frac{9}{22}$; ②$P(B|A_1) = \frac{2}{5}$; ③ The event B is independent of the event $A_1$; ④ $A_1$, $A_2$, and $A_3$ are mutually exclusive events. Write all the correct conclusions' numbers.

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