
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  $n$  be a positive integer. A sequence  $(a, b, c)$  of  $a, b, c \in \{1, 2, . . . , 2n\}$  is called *joke* if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences  $(4, 5, 3)$  and  $(3, 8, 3)$  are jokes, but  $(3, 2, 7)$  and  $(3, 8, 8)$  are not. Determine the number of joke sequences in terms of  $n$ .

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