
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  $p$  and  $q$  be prime numbers. The sequence  $(x_n)$  is defined by  $x_1 = 1$ ,  $x_2 = p$  and  $x_{n+1} = px_n - qx_{n-1}$  for all  $n \geq 2$ .
Given that there is some  $k$  such that  $x_{3k} = -3$ , find  $p$  and  $q$ .

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