Number of rainy days probability distribution in a two-state markov chain

Question

I am working on a Markov Chain model with two states, sunny and rainy.

The transition matrix represents the probabilities of transitioning from sunny to rainy (J) and staying in the rainy state (R) :

\begin{bmatrix} 1-J & J \\ 1-R & R \end{bmatrix}

I'm assuming that I start from a sunny state, so my initial-state vector is :

\begin{bmatrix} 1 & 0 \end{bmatrix}

I'm trying to find the probability that the number of rainy days in a year (365 days) is equal to k for any k.

P(Number of Rainy Days = k) = ?

I tried finding the stationary distribution and then applying Bernoulli's formula, but I'm interested in calculating this probability distribution before equilibrium, hopefully through finding a closed formula.

I appreciate your insights.