
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  $1, 2, . . . , 10$  are randomly arranged in a circle. Let  $p$  be the probability that for every positive integer  $k < 10$ , there exists an integer  $k' > k$  such that 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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