
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  $X$  be a set of  $n + 1$  elements,  $n\geq 2$ . Ordered  $n$ -tuples  $(a_1,\ldots,a_n)$  and  $(b_1,\ldots,b_n)$  formed from distinct elements of  $X$  are called*disjoint*if there exist distinct indices  $1\leq i \neq j\leq n$  such that  $a_i = b_j$ . Find the maximal number of pairwise disjoint  $n$ -tuples.

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