
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  $k$  be an integer and  $k > 1$ . Define a sequence  $\{a_n\}$  as follows: $a_0 = 0$ , $a_1 = 1$ , and $a_{n+1} = ka_n + a_{n-1}$  for  $n = 1,2,...$ .

Determine, with proof, all possible  $k$  for which there exist non-negative integers  $l,m (l \not= m)$  and positive integers  $p,q$  such that  $a_l + ka_p = a_m + ka_q$ .

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