
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
There are $5$ red balls, $2$ white balls, and $3$ black balls in bag A. Bag B contains $4$ red balls, $3$ white balls, and $3$ black balls. First, a ball is randomly taken out from bag A and placed into bag B. Let $A_{1}$, $A_{2}$, and $A_{3}$ denote the events of drawing a red ball, white ball, and black ball from bag A, respectively. Then, a ball is randomly drawn from bag B, and let $B$ denote the event of drawing a red ball from bag B. The following conclusions are given:
1. $P(B)= \frac {9}{22}$
2. $P(B|A_{1})= \frac {2}{5}$
3. Event $B$ and event $A_{1}$ are independent
4. $A_{1}$, $A_{2}$, $A_{3}$ are pairwise mutually exclusive events.
Indicate the correct conclusion(s) by listing the corresponding number(s).

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