
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 prime numbers $2, 3, 5, 7, 11, 13, 17,$ and $19$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two numbers whose sum is a multiple of $10$ are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

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