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The analysis of type IIB flux vacua on warped Calabi-Yau orientifolds becomes considerably involved for a large number of complex structure fields. We however show that, for a quadratic flux superpotential, one can devise simplifying schemes which effectively reduce the large number of equations down to a few. This can be achieved by imposing the vanishing of certain flux quanta in the large complex structure regime, and then choosing the remaining quanta to respect the symmetries of the underlying prepotential. One can then implement an algorithm to find large families of flux vacua with a fixed flux tadpole, independently of the number of fields. We illustrate this approach in a Calabi-Yau manifold with 51 complex structure moduli, where several reduction schemes can be implemented in order to explicitly solve the vacuum equations for that sector. Our findings display a flux-tadpole-to-stabilized-moduli ratio that is marginally above the bound proposed by the Tadpole Conjecture, and we discuss several effects that would take us below such a bound. [ABSTRACT FROM AUTHOR]