Sum of a stopped random walk and a modified random walk

Question

I'm stuck with the following problem:

Let $\{\tilde{X}_n\}_n$ and $\{\tilde{Y}_n\}_n$ be two simple symmetric random walks. Let $\{X_n\}_n$ be $\{\tilde{X}_n\}_n$ stopped when it first hits the level 1 and $\{Y_n\}_n$ given by \begin{eqnarray} Y_0=0, \quad Y_n=\sum_{k=1}^n2^{-k}(\tilde{Y}_k-\tilde{Y}_{k-1}). \end{eqnarray} Identify the distribution of lim inf$_n (X_n+Y_n)$ and show that $\{X_n+Y_n\}_n$ is not uniformly integrable.

I tried a bunch if things but none of them took me anyhere, so any help/hint/opinion is welcome. Thanks!