
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 3 red balls and 2 white balls, while jar B contains 4 red balls and 1 white ball. First, a ball is randomly drawn from jar A and placed into jar B, where event $A_1$ represents drawing a red ball from jar A, and event $A_2$ represents drawing a white ball. Then, a ball is drawn at random from jar B, and event B indicates that the ball drawn from jar B is red. The following statements are given:  
① $P(B) = \frac{23}{30}$  
② Event B is independent of event $A_1$.  
③ Events $A_1$ and $A_2$ are mutually exclusive.  
④ The value of $P(B)$ cannot be determined as it depends on which of the events $A_1$ or $A_2$ occurs.  
Identify the correct statement(s).

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