
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  $k>1$  be an integer. The sequence  $a_1,a_2, \cdots$  is defined as: $a_1=1, a_2=k$  and for all  $n>1$  we have:  $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers  $n$  such that  $a_n$  is a power of  $k$ .

*Proposed by Amirhossein Pooya*

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