
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
Integers  $a_1, a_2, \dots a_n$  are different at  $\text{mod n}$ . If  $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$  are also different at  $\text{mod n}$ , we call the ordered  $n$ -tuple  $(a_1, a_2, \dots a_n)$  *lucky*. For which positive integers  $n$ , one can find a lucky  $n$ -tuple?

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