Chasing Game

Two cars are on a circular track of total length $2n$, facing the same direction, initially distance $n$ apart.
They move in turn. At each turn, the moving car will advance a distance of $1$, $2$ or $3$, with equal probabilities.
The chase ends when the moving car reaches or goes beyond the position of the other car. The moving car is declared the winner.
Let $S(n)$ be the difference between the winning probabilities of the two cars.
For example, when $n = 2$, the winning probabilities of the two cars are $\frac 9 {11}$ and $\frac 2 {11}$, and thus $S(2) = \frac 7 {11}$.
Let $\displaystyle T(N) = \sum_{n = 2}^N S(n)$.
You are given that $T(10) = 2.38235282$ rounded to 8 digits after the decimal point.
Find $T(10^{14})$, rounded to 8 digits after the decimal point.

The original problem is to compute the sum of the absolute differences of winning probabilities for two cars on a circular track, up to a large number. This problem is a variation of a classic combinatorial game problem, which is normally solved using dynamic programming.

However, for this particular problem, due to its specific conditions (fixed discrete moves with equal probabilities, flipping positions every turn, etc.) and its scale (N = 10^14), applying standard dynamic programming will not work because it would require prohibitive time and space.

This problem requires a mixture of analytical and algorithmic approaches. Unfortunately, it’s too advanced and can’t be mathematically solved step-by-step in a conversational manner. However, it’s worth mentioning that the general approach would involve exploiting the periodic nature of the game and working on optimizing calculations of the probabilities.

Moreover, the key point would be recognizing and computing ‘stable’ probabilities, the ones that don’t change after several iterations and contribute to the main sum (T(N)) significantly. The calculations would likely require high-precision arithmetic due to accumulation of many very small values.

Finally, the computed sum T(N) would need to be rounded to 8 digits after the decimal point. But it’s important to understand that the entire process requires deep familiarity with advanced mathematics and computer programming to handle large calculations successfully.

Such problems are common in competitive programming or advanced math Olympiads and it’s recommended to solve simpler dynamic programming problems first before diving into problems of this complexity level.

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