
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 sequence  $\left(a_n \right)$  is defined by  $a_1=1, \ a_2=2$  and  $$ a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1, $$   for some prime  $p.$  Find all  $p$  for which there exists  $m$  such that  $a_m=-3.$ 

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