Find $\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)$ where $X,Y$ poisson

Question

Let $X=Y_1+N,Y=Y_2+N$ where $N\sim \text{pois}(\lambda), Y_1\sim \text{pois}(\lambda_1), Y_2\sim \text{pois}(\lambda_2)$ and $N,Y_1,Y_2$ are independent. I'm asked to find $$\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x).$$I have been trying to rewrite this (using that $F_Y(Y)\sim \text{Uniform}(0,1)$) but can't get it to work out. We also have a result that states that $$\lim_{x\uparrow 1}\mathbb{P}(F_Y(Y)>x|F_X(X)>x)=\lim_{x\uparrow 1}\frac{1-2x+C(x,x)}{1-x}$$where $C$ is the copula for $(X,Y)$ i.e. $$F_{(X,Y)}(x,y)=C(F_X(x),F_Y(y)).$$We know from distribution of bivariate poisson that $$F_{(X,Y)}(x,y)=e^{-(\lambda+\lambda_1+\lambda_2)}\sum_{i=0}^{\min(x,y)}\frac{\lambda^i}{i!}\frac{\lambda_1^{x-i}}{(x-i)!}\frac{\lambda_2^{y-i}}{(y-i)!}$$and that $X\sim \text{pois}(\lambda+\lambda_1), Y\sim \text{pois}(\lambda+\lambda_2)$ however I have not been able to find $C$. Help would be appreciated!