Simple random walk on a circle with states: $\{0,1,2,..., N-1\}, \quad N\geq2$
With state $N-1$ adjacent to state $0$ to complete the circle.
Let $X_n$ be a random walk on the circle starting at $X_0=0$.
Transition probabilities are: $$p(0,N-1)=p(N-1,0)=p(k,k-1)=p(k-1,k)=\frac{1}{2}$$ $$k=1,....,N-1$$
Let $T_k$ denote the first time at which the number of distinct states visited equals $k$, so $T_1=0$ and $T_2=1$.
This means $T_N$ denotes the first time that every state has been visited.
Now, I need to show that $X_{T_N}$, which is a random variable, follows a uniform distribution on $\{1,2,...,N-1\}$
The only way I could think of on how to prove this is from using the fact that ${T_N}\geq N-1$ and showing the following, using computer, for any value of $N\geq2$: $$\lim_{n\to t} P^n=\pi$$ where $P$ is the transition matrix, $\pi$ is the steady state/invariant probability distribution with uniform distribution, and $t$ is any integer $\geq N-1$. This is basically using brute force to prove this empirically.
How do I show this using a more intelligent way (i.e. math derivation)? I spent hours trying to think of a way and came out empty handed.
