A total of m white and m black balls are distributed among two urns, with each urn containing m balls. At each stage, a ball is randomly selected from each urn and
the two selected balls are interchanged. Let X_n denote the number of black balls in urn 1 after the nth interchange.
(a) Give the transition probabilities of the Markov chain X_n, n \geq 0.
(b) Without any computations, what do you think are the limiting probabilities
of this chain?
(c) Find the limiting probabilities and show that the stationary chain is time

Attempted solution:
Transition probabilities
P_{i, i+1} = \frac{(m - i)^2}{m^2}

P_{i,i} =\frac{2(m - i)(i)}{m^2}

P_{i, i - 1} = \frac{i^2}{m^2}

I think those are the transition probabilities. Where i is the current number of black balls.

I also know that in order to be time reversible I need the stationary distribution to satisfy: \pi_iP_{ij} = \pi_jP_{ji}, where \pi is the stationary distribution.

Any assistance as to how I can guess the correct form of the solution based on intuition regarding this scenario would be helpful, thanks in advance.