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 \frac{1}{3} 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|.