
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
Determine the number of positive integers that, in their base \( n \) representation, have all different digits and every digit except the leftmost one differs from some digit to its left by \( \pm 1 \). Express the answer as a simple function of \( n \) and provide a proof of your result.

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