
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
Given a set  $S$  of  $n$  variables, a binary operation  $\times$  on  $S$  is called *simple* if it satisfies  $(x \times y) \times z = x \times (y \times z)$  for all  $x,y,z \in S$  and  $x \times y \in \{x,y\}$  for all  $x,y \in S$ .  Given a simple operation  $\times$  on  $S$ , any string of elements in  $S$  can be reduced to a single element, such as  $xyz \to x \times (y \times z)$ .  A string of variables in  $S$  is called*full*if it contains each variable in  $S$  at least once, and two strings are *equivalent* if they evaluate to the same variable regardless of which simple  $\times$  is chosen.  For example  $xxx$ ,  $xx$ , and  $x$  are equivalent, but these are only full if  $n=1$ .  Suppose  $T$  is a set of strings such that any full string is equivalent to exactly one element of  $T$ .  Determine the number of elements of  $T$ .

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