Hitting zero on a ring

HardProbability Stochastic Processes~9m

Points $0, 1, \dots, 99$ are arranged clockwise on a circle. A token starts at point $1$ and each step moves one point clockwise or counter-clockwise with equal probability. What is the expected number of steps until it first reaches point $0$ ?

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Cut the ring at $0$ : it becomes a path $0..100$ where both ends are the target ( $100\equiv0$ ).
Gambler's-ruin expected duration from $k$ with barriers at $0$ and $N$ is $k(N-k)$ ; here $k=1$ , $N=100$ .

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Hitting zero on a ring

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