
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  $ \alpha,\beta$  be real numbers satisfying  $ 1 < \alpha < \beta.$  Find the greatest positive integer  $ r$  having the following property: each of positive integers is colored by one of  $ r$  colors arbitrarily, there always exist two integers  $ x,y$  having the same color such that  $ \alpha\le \frac {x}{y}\le\beta.$ 

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