
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
Pratyya and Payel have a number each,  $n$  and  $m$  respectively, where  $n>m.$  Everyday, Pratyya multiplies his number by  $2$  and then subtracts  $2$  from it, and Payel multiplies his number by  $2$  and then add  $2$  to it. In other words, on the first day their numbers will be  $(2n-2)$  and  $(2m+2)$  respectively. Find minimum integer  $x$  with proof such that if  $n-m\geq x,$  then Pratyya's number will be larger than Payel's number everyday.

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