\begin{align*}
    U_{\tau} & (\max(k,k'))
    \Pr(X_j \leq C_j, 
    Z_{X_j} = i, \min(X_j, C_j) = \max(k,k'), Y_j \leq \min(k,k'))\\
    &=f^{0i}_{*,\tau}(\max(k,k'))
    \Pr(Y_j \leq k \leq \min(X_j, C_j), Y_j \leq k' \leq \min(X_j,C_j)),
\end{align*}
