Under what assumptions $Y_n \xrightarrow{L^1} 0$ as $n \to \infty\ $?

Let $\{X_n, U_n\ |\ n \geq 1 \}$ be a collection of independent random variables such that $X_n$'s are iid random variables following standard Cauchy distribution and $U_n$'s are independent random variables following Bernoulli distribution with parameter $\frac 1 n.$ Define a sequence of random variables $\{Y_n \}_{n \geq 1}$ by $$Y_n = \begin{cases} 0 & \text{if}\ U_n = 0 \\ \frac {X_n} {n} & \text{if}\ U_n = 1 \end{cases}$$

Questions$:$

$(1)$ Does $Y_n \xrightarrow{L^1} 0\ $?

$(2)$ If $(1)$ is false, can we slightly modify the above hypothesis (without altering the distributions) so that $(1)$ holds under the modified hypothesis?

My Attempt$:$$(1)$ is false because of the following reason $:$$$\begin{align*} \mathbb E \left [|Y_n| \right ] & = \mathbb E \left [|Y_n|\ \lvert\ U_n = 0 \right ] \mathbb P (U_n = 0) + \mathbb E \left [|Y_n|\ \lvert\ U_n = 1 \right ] \mathbb P (U_n = 1) \\ & = \frac {\mathbb E \left [|X_n| \right ]} {n^2} \\ & = \infty\end{align*}$$ as $\mathbb E [|X_n|] = \infty$ for all $n \geq 1.$ But what about $(2)\ $? Do anyone have any idea about it?

Regards

Anil