
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 \( x y z \rightarrow x \times (y \times z) \). 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, \( x x x, x x \), and \( x \) are equivalent, but these are only full if \( n=1 \). Suppose \( T \) is a set of full 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}
user