Splay Trees

The number $\frac{1}{2}$ is written on a blackboard. For a real number $c$ with $0 < c < 1$ , a $c$ -splay is an operation in which every number $x$ on the board is erased and replaced by the two numbers $cx$ and $1-c(1-x)$ . A splay-sequence $C = (c_1,c_2,c_3,c_4)$ is an application of a $c_i$ -splay for $i=1,2,3,4$ in that order, and its power is defined by $P(C) = c_1c_2c_3c_4$ .

Let $S$ be the set of splay-sequences which yield the numbers $\frac{1}{257}, \frac{2}{257}, \dots, \frac{256}{256}$ on the blackboard in some order. If $\sum_{C \in S} P(C) = \tfrac mn$ for relatively prime positive integers $m$ and $n$ , compute $100m+n$ .