Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his score. A player wins the game if their score is higher than the other player's score. 

The probability that Dukejukem defeats Trollner to win WASS equals $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Compute $m+n$.

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