probability - Where am I wrong in the following strategy?

I have two types of variables $X_{l,m}$ and $Y_m$. All $X_{l,m}$ are i.i.d distributed according to some probability distribution. Similarly all $Y_m$ are i.i.d distributed according to some other probability distribution. I want to find the following probability $$P\left(\max_{l=1 \cdots L}\left\{\min_{m=1\cdots M}\left(X_{l,m}+Y_m\right)\right\}

My Strategy:

1- First find the PDF and CDF of a generic sum $Z=X+Y$.

2- Find the PDF and CDF of

$$T=\min_{m=1\cdots M}Z_m$$

3- Take the $L$th power of the CDF of $T$ to find the overall probability.

Is there something fundamentally wrong with this strategy? Please clarify if there is something wrong. On the other hand if my strategy is right then how can I verify it. Many many thanks in advance.

Second Strategy:

In this strategy I will fix values of all $Y_m$ and find the conditional probability and then average it over all $Y_m$'s. Obviously I will have to take the limits of at least one of the $Y_m$'s from $0-->C$. But this strategy seems to be much more difficult then the first one. Any comments on my strategies will be highly appreciated. Thanks in advance.