
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
Yang has a standard  $6$ -sided die, a standard  $8$ -sided die, and a standard  $10$ -sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as  $\frac{m}{n}$ , for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by Yannick Yao*

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