
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 ($a_1$, $a_2$, ... $a_{12}$) be a list of the first 12 positive integers such that for each $2 \leq i \leq 12$, exactly one of either $a_i + 1$ or $a_i-1$ (but not both) must appear somewhere before $a_i$ in the list. Determine the total number of such lists.

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