Six chairs are arranged in a row, and six people randomly choose a seat among these chairs. Each person then decides one of three positions for their legs:  
1. Feet on the floor  
2. Legs crossed to the right  
3. Legs crossed to the left  

A problem arises only if two adjacent people have the following leg positions:  
- The person on the right crosses their legs to the left.  
- The person on the left crosses their legs to the right.

Determine the probability that this situation does **not** occur. Express the probability as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the sum $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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