
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, white, and black ball, respectively. Then, randomly take a ball from jar B, and let event $B$ represent 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. $P(B)= \frac {2}{5}$;
2. $P(B|A_{1})= \frac {5}{11}$;
3. Event $B$ and event $A_{1}$ are independent;
4. $A_{1}$, $A_{2}$, $A_{3}$ are mutually exclusive events.

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