
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
If $X$ is a set, and $\tau$ is a set consisting of some subsets of $X$, and it satisfies: ① $X$ belongs to $\tau$, $\emptyset$ belongs to $\tau$; ② The union of any number of elements in $\tau$ belongs to $\tau$; ③ The intersection of any number of elements in $\tau$ belongs to $\tau$. Then $\tau$ is called a topology on the set $X$. Given the set $X=\{a, b, c\}$, for the following four sets $\tau$:  
① $\tau=\{\emptyset, \{a\}, \{c\}, \{a, b, c\}\}$;  
② $\tau=\{\emptyset, \{b\}, \{c\}, \{b, c\}, \{a, b, c\}\}$;  
③ $\tau=\{\emptyset, \{a\}, \{a, b\}, \{a, c\}\}$;  
④ $\tau=\{\emptyset, \{a, c\}, \{b, c\}, \{c\}, \{a, b, c\}\}$.  
Among these, the sequence numbers of the sets $\tau$ that are topologies on the set $X$ are     .

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