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We introduce a novel regression model for the conditional left and right tail of a possibly heavy-tailed response. The proposed model can be used to learn the effect of covariates on an extreme value setting via a Lasso-type specification based on a Lagrangian restriction. Our model can be used to track if some covariates are significant for the lower values, but not for the (right) tail---and vice-versa; in addition to this, the proposed model bypasses the need for conditional threshold selection in an extreme value theory framework. We assess the finite-sample performance of the proposed methods through a simulation study that reveals that our method recovers the true conditional distribution over a variety of simulation scenarios, along with being accurate on variable selection. Rainfall data are used to showcase how the proposed method can learn to distinguish between key drivers of moderate rainfall, against those of extreme rainfall.