
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  $ \{ a_n \} _ { n \ge 0 } $  is defined by  $ a_0 = 2 , a_1 = 4 $  and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers  $ n $ . Determine all prime numbers  $ p $  for which there exists a positive integer  $ m $  such that  $ p $  divides the number  $ a_m - 1 $ .

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