
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
The integers  $a_0, a_1, a_2, a_3,\ldots$  are defined as follows: $a_0 = 1$ ,  $a_1 = 3$ , and  $a_{n+1} = a_n + a_{n-1}$  for all  $n \ge 1$ .
Find all integers  $n \ge 1$  for which  $na_{n+1} + a_n$  and  $na_n + a_{n-1}$  share a common factor greater than  $1$ .

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