Conditional probability with poisson?

I'm reviewing for a final exam, and I got stuck on one of the practice problems, and was hoping someone could point me in the right direction:

The question is: given , where and are independant Poissons with different means, find

So to do this, I figure I'd need to find , but I'm having trouble getting .

I have: , which I can use independance (of U & W) to separate and get but I think that would actually be , and I'm not sure marginalizing over v to get is the right approach.

Could someone tell me if this is the right direction to head in, or if there's a better/easier way to get ?

thanks!