Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b  \times 5^c  \times 7^d  \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.

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