Find distribution of Random Variable $X_N$ where $N$ is independent random variable, $\sum\limits_{j=1}^n P(N=j)=1$ & each $X_i~exp(\lambda_i)$

Question

Question : Let $X_1,X_2,\ldots,X_n$ be independent random variables such that $X_i\sim\exp(\lambda_i)$ such that if $i\neq j$ then $\lambda_i\neq\lambda_j$.

Let $N$ be independent of these random variables such that $\sum\limits_{j=1}^n P(N=j) = 1$

Find density, CDF, and hazard function of random variable $X_N$.

My Approach
Okay so firstly I'm not asking for help on CDF and hazard function, I'm wondering whether or not I'm able to simplify the equation I derived using total probability formula further.

I got $$ P_{X_N}(t) = \sum_{j=1}^n \lambda_j e^{-\lambda_j t} P(N=j) $$ Seems to me I can't go any further without knowing the actual distribution of $N$, however I want to make sure I can't go further just in case.