
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 function $f(x) = \left[ x \left[ x \right] \right]$, where $\left[ x \right]$ denotes the greatest integer not greater than $x$, when $x \in (0, n]$, $n \in \mathbb{N}^*$, the range of the function $f(x)$ is the set $A_n$. The number of topologies $\tau$ on the set $A_2$ that contain 4 elements is \_\_\_\_\_\_.

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