
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  $S_0 = \varnothing$  denote the empty set, and define  $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$  for every positive integer  $n$ .  Find the number of elements in the set
\[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \]

*Proposed by Evan Chen*

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