
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
Let $A$ and $B$ be two random events. If ① $A$ and $B$ are mutually exclusive events with $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{2}$, then $P(A\cup B)=\frac{1}{6}$; ② if $A$ and $B$ are complementary events, then $P(A\cup B)=1$; ③ if $A$ and $B$ are independent events with $P(A)=\frac{1}{3}$ and $P(B)=\frac{2}{3}$, then $P(A\overline{B})=\frac{1}{9}$; ④ if $P(\overline{A})=\frac{1}{3}$, $P(\overline{B})=\frac{1}{4}$, and $P(\overline{A}B)=\frac{1}{4}$, then $A$ and $B$ are independent events. The correct numbers of the above propositions are ______. (Fill in the numbers)

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