
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
For any integer  $n \ge2$ , we define  $ A_n$  to be the number of positive integers  $ m$  with the following property: the distance from  $n$  to the nearest multiple of  $m$  is equal to the distance from  $n^3$  to the nearest multiple of  $ m$ . Find all integers  $n \ge 2 $  for which  $ A_n$  is odd. (Note: The distance between two integers  $ a$  and  $b$  is defined as  $|a -b|$ .)

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