Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that 
\begin{align}
\mathbb{E} [ W^2 (Z^2 - 1)] = 2 a^2.
\end{align}
Now suppose I truncate random variable $W$.
I was wondering what truncation level $R$ should be such that 
\begin{align}
\mathbb{E} [ W^2 \cdot  \mathbf{1}\{ | W| > R\} \cdot  (Z^2 - 1)] \leq  a^2.
\end{align}
Can we set $R$ to be a constant?
This seems an easy problem, but computing the expectation seems not trivial.