
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  $x_0=a,x_1=b$  and  $x_{n+1}=2x_n-9x_{n-1}$  for each  $n\in\mathbb N$ , where  $a,b$  are integers. Find the necessary and sufficient condition on  $a$  and  $b$  for the existence of an  $x_n$  which is a multiple of  $7$ .

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