
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
The numbers \(1, 2, \ldots, 10\) are randomly arranged in a circle. Let \(p\) be the probability that for every positive integer \(k < 10\), there exists an integer \(k'\) such that \(k' > k\) and there is at most one number between \(k\) and \(k'\) in the circle. If \(p\) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \(a\) and \(b\), compute \(100a + b\).

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