
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 Jar A, and 4 red balls, 3 white balls, and 3 black balls in Jar B. First, a ball is randomly drawn from Jar A and put into Jar B. Let events $A_{1}, A_{2},$ and $A_{3}$ represent that the ball drawn from Jar A is a red ball, a white ball, and a black ball, respectively. Then, a ball is randomly drawn from Jar B, and let event $B$ represent that the ball drawn from Jar B is a red ball. Among the following conclusions, the correct one(s) is(are) \_\_\_\_\_\_ (write down the number of all correct conclusions).

$① P(B)= \frac {2}{5};$  
$② P(B|A_{1})= \frac {5}{11};$  
$③$ Events $B$ and $A_{1}$ are mutually independent;  
$④ A_{1}, A_{2}, A_{3}$ are pairwise mutually exclusive events;  
$⑤$ The value of $P(B)$ cannot be determined because it depends on which one of the events $A_{1}, A_{2}, A_{3}$ occurs.

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