
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  $V_0 = \varnothing$  be the empty set and recursively define  $V_{n+1}$  to be the set of all  $2^{|V_n|}$  subsets of  $V_n$  for each  $n=0,1,\dots$ . For example \[
	V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\}
	\quad\text{and}\quad
	V_3 
	=
	\left\{ 
	  \varnothing,
	  \left\{ \varnothing \right\},
	  \left\{ \left\{ \varnothing \right\} \right\},
	  \left\{ \varnothing, \left\{ \varnothing \right\} \right\}
	\right\}.
\] A set  $x \in V_5$  is called *transitive* if each element of  $x$  is a subset of  $x$ . How many such transitive sets are there?

*Proposed by Evan Chen*

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