
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
Determine all pairs  $(n,m)$  of positive integers for which there exists an infinite sequence  $\{x_k\}$  of  $0$ 's and  $1$ 's with the properties that if  $x_i=0$  then  $x_{i+m}=1$  and if  $x_i = 1$  then  $x_{i+n} = 0.$ 

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