A Liouvillian extension is a differential field built up from rational functions by taking indefinite integrals, exponentials and algebraic extensions. In 1991, Singer proposed an algorithm to find all in-field solutions of any linear differential equation with coefficients in a given Liouvillian extension. A building block of this algorithm is a loop aiming to find a degree bound of polynomial solutions. Singer asked whether there is an explicit formula to describe the number of iterations in the loop in terms of the order of the differential equation. The question remains unanswered for more than three decades. In this talk, we introduce a new viewpoint to Singer's algorithm, called auxiliary reduction, and provide a tight bound for the number of the iterations.