
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
Assume  $n$  is a positive integer. Considers sequences  $a_0, a_1, \ldots, a_n$  for which  $a_i \in \{1, 2, \ldots , n\}$  for all  $i$  and  $a_n = a_0$ . 

(a) Suppose  $n$  is odd. Find the number of such sequences if  $a_i - a_{i-1} \not \equiv i \pmod{n}$  for all  $i = 1, 2, \ldots, n$ .

(b) Suppose  $n$  is an odd prime. Find the number of such sequences if  $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$  for all  $i = 1, 2, \ldots, n$ .

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